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.

Two Sides of the Same Coin:
Transformations, Functions, and the Common Core

### Presentation

Scott Steketee and Daniel Scher presented a session at NCTM's 2014 Annual Meeting describing Sketchpad-based activities to help build student understanding of transformations in geometry and of functions in algebra by taking advantage of the fact that both are instances of the same mathematical concept.

The handout, presentation sketch, and other resources are available below.

Below are interactive glimpses of the first two activities from the presentation. (Each downloadable activity includes a sketch, a student worksheet, teacher notes, and an explanatory video.) In the first activity, students drag each of the independent variables, observe function behavior, and figure out how which functions show similar behavior and which is different. In the second activity, students apply a function to an independent point, use function notation to label the output, drag to explore the relative rate of change, and restrict the domain to a polygon.

### Overview

In the realm of geometry, the presenters showed examples in which students identify functions and function families, use function notation in a meaningful way, restrict an independent variable to a specific domain, and experience in a concrete tactile way the rate of change of several functions belonging to different families.

As a transition from the geometric realm to the algebraic, the presentation included examples in which students restrict geometric point variables to a number line, make the connection between particular families of geometric transformations and the algebraic operations of multiplication and addition, and explore function behavior and rate of change using dynagraphs.[1]

In the algebraic realm, the presentation included an example in which a function's dynagraph representation is converted to a Cartesian representation by rotating the output axis and generating the Cartesian graph by tracing while dragging the independent variable. In a second example, this combined representation is used to represent the transformation of a parent function by stretching and translating in both horizontal and vertical directions, giving students a clear visual picture of what is often a confusing process for them. In a third example, students play a game in which they experience a function's rate of change by trying to follow the motion of the function's dependent variable.

Finally, an algebraic function was brought back to the geometric realm by applying a sine function to a picture, resulting in a visual special effect. (This example appears in the YouTube video "Wavy Bridge.")

### Background

Common Core Standard G-CO2 says that students should “describe transformations as functions that take points in the plane as inputs and give other points as outputs.”

The fact that geometric transformations are functions creates interesting and enjoyable opportunities to use this connection to help students understand fundamental function concepts.

By providing students with early kinesthetic and visual experiences with geometric transformations (reflections, rotations, translations) we can help students develop deeper and more robust function concepts and create opportunities for explicit discourse about important function ideas, and at the same time expose students to some beautiful connections between geometry and algebra.

### Transformations

Beginning in grade 8 the Common Core uses transformations as the bedrock on which congruence and similarity are built. In high school, the role of transformations is central not only to congruence and similarity, but also to properties of triangles and polygons (such as the base angle theorem for isosceles triangles). Similarity too plays a fundamental role, in defining the sine, cosine, and tangent for the acute angles of a right triangle.

### Functions

Also beginning in grade 8, functions get their own distinct major CCSM category in addition to other Algebra topics. It would be hard to overstate the importance of function in students’ mathematical experiences, yet we also know that students struggle with many function concepts. A number of the difficulties students have with functions can be alleviated by exploiting the connections between functions in geometry and functions in algebra. These activities can promote understanding of variables, function families, function notation, domain and range, composition, rate of change, and much more.

### Ongoing

We encourage you to use these activities with your students, and we’d love to get your feedback to help us refine the activities. As we develop more activities and other support materials, we’ll make them available on www.geometricfunctions.org.

### Note

[1] The activities shown were supported in part by the National Science Foundation Dynamic Number grant, award #0918733. Opinions and views are the presenter’s, and not necessarily those of the NSF.

[2] A dynagraph represents a function using two parallel axes, one for the independent variable and one for the dependent variable. Students investigate function behavior by dragging the independent variable and observing the motion of the dependent variable. (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.)