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:

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NCTM 2015 Regional Meeting, Session 245

Geometric Transformations and Linear Functions: Two Sides of a Coin

Summary: In this presentation, Scott Steketee and Daniel Scher show a sequence of geometric functions activities. In these activities 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.

Link to the presentation web sketch.

Activities

The presentation involved six activities. When used with students, each activity is expected to take one or two class periods. Each activity is accompanied by a student worksheet.

1. The Dilate Family: This activity comes from the introductory unit on Functions and Function Families, and is included here because it provides students with important prerequisite understandings about geometric functions and gives them practice working with Web Sketchpad tools.
2. Reduce the Dimension: Students begin with the already-familiar geometric functions in Flatland and figure out how to move each function into Lineland by restricting the domain of the independent variable to a line and adjusting the function so that the range of the dependent variable lies on the same line.
3. Number the Domain: Once the functions are in LineLand, students turn the restricted domain into a number line, measure the values of the independent and dependent variables, and figure out the numeric operation that corresponds to each geometric function.
4. Compose on a Line: Students compose two Flatland functions (the two that correspond to multiplication and addition), again restricting the domain to a number line, and examine the behavior of the resulting function both geometrically and numerically.
5. Create a Dynagraph: Students shift the output variable vertically to create a Dynagraph representation of their linear function. They drag the independent variable to study the behavior of various linear functions and to solve puzzles in which they adjust the scale factor (for dilation) and the vector (for translation) to match given mystery functions.
6. Connect to Cartesian: Students rotate the output variable of their composed dilation/translation function by 90° and use the resulting Cartesian axes to create and analyze their function’s Cartesian graph, and to solve puzzles in which they adjust the original dilation and translation functions to match given mystery graphs.

Objectives

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

• 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
• Figure out how to move 2D geometric functions onto a 1D number line
• Analyze the numeric behavior of the resulting 1D number-line functions
• Relate the 2D behavior of the original geometric functions to the corresponding numeric behavior of the number-line functions
• Create dynagraph representations of number-line functions
• Use dynagraphs to analyze and explain the behavior of linear functions
• Solve mystery-function dynagraph puzzles by matching the behavior of a constructed function to the observed behavior of a mystery function
• Create perpendicular-axis representations of number-line functions
• Use perpendicular-axis representations to construct Cartesian graphs
• Solve mystery-function Cartesian puzzles by matching the shape of a constructed graph to the observed shape of a mystery function’s graph

Requirements

These activities require web access using a browser that supports HTML5 and JavaScript. (That means almost any current browser.)

Update History:
30 October 2015: Revised based on the presentation at the Atlantic City Regional.
15 April 2015: Derived this page from the original Cartesian Connection page, to support the presentation at the 2015 NCTM Annual Meeting.

* Dynagraphs were invented by Paul Goldenberg, Philip Lewis, and James O’Keefe. See their classic 1992 article “Dynamic representation and the development of a process understanding of function.” In G. Harel and E. Dubinsky (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy. MAA Notes V. 25 (1992), Mathematical Association of America.