Forging Connections
Put Variables that Really Vary Back into the Graph

ATMOPAV Fall Conference, November 9, 2019

Introduction (5 min)

Donovan Hayes, Hill-Freedman World Academy, Philadelphia,

Scott Steketee, 21st Century Partnership for STEM Education,


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.


  1. Big Ideas (5 min)
  2. Unit 1: Functions and Function Families (Quick Tour, 10 minutes)
  3. From Flatland to Numberland (Restricting the Domain, 5 min)
  4. Construct a Dynagraph (Pass the Mouse, 15 min)
  5. Create a Cartesian Graph (Short Demo, 5 min)
  6. Cartesian Games (Concreteness Fading, 10 min)
  7. 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.








Functions and Families (10 min)


The Reflect Family lesson is located at

Function Dances

These students are connecting reflection, rotation, translation, and glide reflection to dance moves.

Flatland to Numberland (5 min)

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

Construct a Dynagraph (15 min)

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)

In the capstone of this series of lessons, 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.

Cartesian Games (10 min)

This sequence of four games challenges students while at the same time employing concreteness fading to develop their understanding of Cartesian graphs in a way that remains connected with the variations that's implicit in the graph. The first challenge is a game based on the actual moving variables, but by the final game the movement is no longer explicit, but remains as a fundamental part of students’ understanding.

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?

▹︎︎ The Fine Print


These lessons 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

[an error occurred while processing this directive]

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.