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.

Graph Dancer Games

In these games students experience the rate of change of a function by trying to move the dependent variable at the correct rate while the independent variable moves at a constant speed.

Description

In the Graph Dancer Sketchpad games, students use their mouse or
finger to drag a character on the screen in a way that matches the
behavior of a mathematical function. These dances can provide
students with engaging sensory-motor experiences of the variation
that characterizes any function taking a real number as input and
giving a real number as output. To dance well, the dancer must
practice, and develop a feel for the dynamic behavior of the
function. When must she move the character up, and when down?
Where will she encounter a relative minimum or maximum? Where must
she dance more quickly, and where must she slow down or even come
to a stop? Through these dance moves, the dancer physically
experiences increasing and decreasing functions and greater and
lesser relative rate of change of *y* with respect to *x.*
Every function's dance has its own feel, though functions from the
same family have dances with similar patterns and similar rhythms.
(These games are the real-number analog of the Dance the Dependent
Variable--Geometric Function Dances activity from the Dynamic
Number collection.)

Each game begins with a three-second countdown to allow the
"dancer" to prepare, following which the *x*-value begins
moving at a constant rate along a restricted domain while the
dancer drags a traced point in time with the *x* value. As
she does so, the dancer experiences the function behavior through
her motor system: to avoid going ahead or lagging behind she must
attend not only to the *y*-value of the function but also
to the varying rate of change of *y* with respect to *x*.
The details of the dances fall into two broad categories, and
differ within each category based on the hints the dancer has
available to stay on track.

Games 1 and 2 are one-dimensional dances. They require the
dancer to drag the traced point in just a single dimension, up and
down along the *y*-axis, trying to follow the correct *y*-value
based on appropriate hints. In these games the dancer concentrates
only on the *y* value's behavior, trying take on, through
dragging, the precise dynamic behavior of the dependent variable.

Games 4 and 5 are two-dimensional dances. They require the
dancer to move in two dimensions on the Cartesian plane,
simultaneously keeping pace with the constant horizontal velocity
of *x* and with the varying vertical velocity of *y*.
Though these dances follow the path of the graph (which may or may
not be a visible hint, depending on the game), the dancer must
follow not only the shape of the graph but also its dynamics. The
dancer must maintain the correct constant horizontal (*x*)
rate while changing the vertical (*y*) rate according to
the function's behavior.

Game 3 is a transition dance, with the traced point being moved automatically in the horizontal direction but under the dancer's control in the vertical direction.

[In Games 1 and 2 the dancer corresponds to a real number, and a correct dance corresponds to the output of the function: the student is dancing the role of dependent variable. In Games 4 and 5 the dancer corresponds to an ordered pair, and a correct dance matches the location and speed of both the input and the output: the student is dancing the role of the ordered pair that makes up the graph.]

Status

These games are in prototype form. At the present time, no student worksheets or teacher notes are available. Only the Sine Dancer sketch exists, with the following five games:

- In Dance 1 the dancer must follow a blue dot (the true
*y*value) as it goes up and down the*y*-axis. A traced point tracks the animated*x*value and the dancer's*y*value to trace an approximation of the function's graph. - In Dance 2 the blue dot is missing. Instead, the function's graph is visible and the dancer must use the graph to decide how to move. An accurate dance results in a trace that's very close to the displayed graph..
- In Dance 3 the player drags vertically to match the y-value of the dot on the y-axis, while the dragged character automatically keeps pace with the x-value horizontally.
- In Dance 4 the player must move along the graph with a
constant horizontal velocity and a vertical velocity that
corresponds to the relative rate of change of
*y*with respect to*x*. - In Dance 5 the blue dot is back! The player must move
horizontally to keep pace with the
*x*dot (on the*x*axis) and vertically to keep pace with the*y*dot (on the*y*axis). There is no graph to follow.

Feedback

We need feedback from teachers and/or parents who try these games with their students. Are all five games useful? Which are most important, and which least? How can the sketches be refined to make them easier to understand, more convenient to use, etc.? We'd also welcome suggestions for the student worksheet and for the teacher notes.

Sine Dancer

This is the only sketch available at this time. But it uses Sketchpad, so you can edit the function on any of the five game pages to see what it feels like to dance a linear function, a quadratic, a cubic, an exponential or logarithmic function, an absolute value function, a rational function, and so forth. The download currently contains only a sketch; the student worksheet and teacher notes remain to be written.

Earlier Prototypes

The first prototype had only two game sketches called Dance 1 and Dance 2. Each sketch contains pages for a variety of functions: a linear function, a piecewise linear function, and so forth, with the last page using a piecewise function with two different types of discontinuities.

**Graph Dancer Part 1: **In
Part 1 your hint is a blue dot moving up and down on the y-axis,
so all you have to do is dance up and down along with it.
(Mathematically, the hint is the actual y-value and that's what
you need to match with your dragging. Thus you are varying *y*
in order to create an accurate Cartesian graph.)

**Graph Dancer Part 2:**
In Part 2 the only hint is the graph as you move the arrow up and
down the y-axis. (Mathematically, the hint is the Cartesian graph
that would result from accurate variation of y. Thus you are
working backward from the Cartesian graph to vary *y*
accurately.)

04 January 2014: Incorporated this page into the new website.

31 October 2013: posted original versions of this page and the Sine Dancer sketch.