Technologically Embodied
Geometric Functions
Introduce Function Concepts and Linear Functions Geometrically(!) with Web Sketchpad

▿ Overview


CCSSM expects students to understand transformations as functions. With Web Sketchpad, algebra students exploit this standard: Using geometric transformations they vary variables; experience domain, range, and rate of change; and connect their learning to the graph of `y = mx + b`.

Richard Feynman famously said “What I cannot create, I do not understand.” The focus of today’s session is to enable students to create their own functions with variables that students actually vary, with relative rate of change that they observe, analyze, and control, and with functions whose behavior creates patterns and pictures. These sensorimotor experiences of creating, manipulating, and observing mathematical objects are all aimed at helping students develop a deep and broad understanding of function concepts.

▹︎ Agenda

  1. Big Ideas (5 min)
  2. The Dilate Function (Participants, 15 min)
  3. Restrict the Domain (5 min)
  4. Create a Dynagraph (Participants, 20 min)
  5. Create a Cartesian Graph (5 min)
  6. Discuss, Reflect, Q&A (Participants, 10 min)

▹︎ Big Ideas (5 min)

As students construct, manipulate, and analyze transformations from a function point of view, they build conceptual understanding through a focus on the big ideas.






▹︎ The Dilate Function (15 min)


Download the student worksheet.      Video Step 2       Video Step 7       Play the Dilate Games.      Video Alice Shrinks and Grows

In this activity you'll create an independent variable, a rule, and a dependent variable. You'll vary the variables, make some patterns, analyze the relative rate of change, and solve interesting challenges.

▹︎ Restrict the Domain (5 min)


Download the student worksheet.      Video Step 2       Video Step 3       Video Question 6

What happens when students restrict the domain of the Dilate and Translate function families to a number line, and then compose these two functions?

▹︎ Create a Dynagraph (20 min)


Download the student worksheet.      Video Step 2       Video Step 4       Video Step 6       Video Step 10       Play the Dynagraph Game.      Video Movie Trailer

Create your own composed Dilate-and-Translate function restricted to a number line, modified to put the dependent variable on a second number line. Analyze how the dilation scale factor and translation vector affect the motion of the points, and how they relate the values of the variables. Finish by playing the Dynagraph game, trying to match the behavior of a mystery function.

▹︎ Create a Cartesian Graph (5 min)


Download the student worksheet.      Video Step 2       Video Step 3       Video Step 6       Video Step 7       Play the Cartesian Games.

In the capstone of this series of activities, students create the Cartesian graph of a linear function using only dilation, translation, and rotation functions, and understand the shape of the resulting graph in terms of the dilation scale factor and the translation vector.

▹︎ Discuss & Reflect (10 min)

How is this approach in keeping with Feynman’s observation about creating in order to understand?

How might it work in your own classroom?

What problems would you anticipate?

How might it encourage your students to notice, and to wonder?

What other questions and comments do you have?

▹︎ Materials

A sequence of free classroom-ready activities implementing this approach is available on this web site.

The activities are divided into two units. Click on either heading below to go to that unit’s web page.

  1. Investigate Geometric Transformations as Functions 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.
  2. Connect Geometry and Algebra Through Functions connects the geometry of transformations to linear functions. 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 geoemetrically-constructed functions in both Dynagraph and Cartesian graph forms.

The activities in both units include dynamic websketches, student worksheets that can be viewed online or downloaded for printing, and help videos that scaffold student’s use of Web Sketchpad tools.

▹︎ The End

Presentation Web Page:

Mathematics Teacher Article:

▹︎︎ The Fine Print


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

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Update History:

10 January 2017: Created this page for a workshop at the Philadelphia Education Fund.
01 March 2017: Revised to use a new format for hiding/showing sections of a page.