Technologically Embodied

Geometric Functions

Geometric Functions

About this website

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.

Connect Geometry and Algebra Through Functions

▿ Overview

This unit, featured in the February 2016 issue of *Mathematics Teacher,* provides a transition from functions in geometry to
functions in algebra. (Readers of the online version of the article will note that all figures after the first are live websketches: figures that can be directly manipulated on the article’s web page.)

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. In the culminating activities students represent these geometrically-constructed functions in both Dynagraph* and Cartesian graph forms.

▹︎ Video

The video below shows and describes the activities of this unit in less than 7 minutes.

▹ Status

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, though we hope to be able to provide those soon. We are revising the worksheets, websketches, and assessment materials as we we continue using them with students and observing students' successes and challenges. Please contact the webmaster with your questions, comments, and suggestions for improvements, or for information about our field-test program.

▹ Activities

Each activity below is expected to require one or two class periods. The first activity is a review activity from the introductory unit to prepare students for the rest of the activities. Click any blue header below to go to that activity's web page.

- The Dilate Family: This activity comes from the Introducing Transformations unit, and is included here so that students can review important prerequisite understandings about geometric functions and so they can practice working with Web Sketchpad tools.
- 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.
- 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.
- Compose on a Line: Using a number line as a restricted domain, students compose two geometric functions and examine the behavior of the resulting composed function both geometrically and numerically.
- 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.
- 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.
- Cartesian Games: Students play a series of games designed to help them understand the Cartesian graph of a linear function in terms of the relative rate of change of the independent and dependent variables. In each game, students control scale factor
*s*and translation vector*v*to adjust function*T*(*D*(*x*)) so that it matches mystery function*??*(*x*). Each successive game moves a bit further in the direction of abstraction. The first major step is fading the role of the dependent variables and encouraging students to use instead the plotted points (*x*,*T*(*D*(*x*))) and (*x*,*??*(*x*)), and the second major step is to move toward using the graph itself, and to extract from the shape and location of the graph the needed information for adjusting*s*and*v*.

▹ 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.

▹︎︎ 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**

2015Q4Update1, Semantic Version: 4.5.0-alpha, Build Number: 1013, Build Stamp: ip-10-149-70-76/20160303102907

Web Sketchpad Copyright © 2015 KCP Technologies, a McGraw-Hill Education Company.

“You may freely use Web Sketchpad in your own Web pages, provided you do so for non-commercial use only” [per kcpt.github.io].

“You may freely use Web Sketchpad in your own Web pages, provided you do so for non-commercial use only” [per kcpt.github.io].

Activity License: Creative Commons CC-BY-NC-SA 4.0

**Update History:**

01 February 2016: Updated to link to the newly published MT article, and added the introductory video that summarizes the unit’s activities.

29 July 2015: Completed missing worksheets & answer forms; reformatted the activity web pages.

11 June 2015: Updated links to recently drafted activities.

28 March 2015: Revised to include the Number the Domain activity; simplified the introductory section; and added objectives

21 March 2015: Expanded the descriptions of the activities.

11 August 2014: Created this page.

29 July 2015: Completed missing worksheets & answer forms; reformatted the activity web pages.

11 June 2015: Updated links to recently drafted activities.

28 March 2015: Revised to include the Number the Domain activity; simplified the introductory section; and added objectives

21 March 2015: Expanded the descriptions of the activities.

11 August 2014: Created this page.

* 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.