**Introduction**

Function is a
fundamental concept in mathematics; it is one that students explore repeatedly,
at increasing levels of sophistication, throughout the early and middle grades
(and beyond) (Steketee and Scher 2011). In the early grades, students may
encounter functions as lists of inputs and outputs in classroom activities such
as “Guess My Rule” (Huinker
2002). With that activity, students are given input values one by one as
suggested in **figure 1**. With each
additional input, students are asked to construct a *single* rule that transforms each value in the “input” column into
the corresponding “output” value at right.

**Fig.
1 **Students record their data. (From “Guess
My Rule,” Huinker 2002, p. 320)

In subsequent activities, students may consider functions
as “machines” that provide a specific output for a given input (Reeves 2005, p.
251). For instance, the tasks in **figure
2** were developed for third graders studying functions.

**Fig.
2** These illustrations render a third-grade
version of a function machine task.

In the middle grades (and beyond), students experience
functions as (a) plots of ordered pairs on the coordinate plane and (b) as
formulas to be evaluated for particular values of *x *(King 2002). Such representations are highlighted in **figure 3**.

**Fig. 3 **Graphical, tabular, and symbolic representations of are available at http://bit.ly/demos-graph__.__

Unfortunately, none of these representations adequately
capture the *dynamic* nature of
function; instead, each “portray(s) input and output in a discrete and static way”
(Steketee and Scher 2011, p. 49).

**Dynagraphs:
An Introduction**

As an alternative to function representations presented in
**figures 1–3**, consider the dynagraph
sketch shown in **figure 4**.

**Fig.
4** This dynagraph example is available at http://bit.ly/dynagraph1__. __

The point along the bottom number line represents an *input value *of an unknown function. The
upper point represents the corresponding *output
value* of the function for the given input. As the bottom point is dragged,
students make conjectures concerning the relationship between the input and
output. For instance, looking at a dynagraph, such as that depicted in **figure 4**, students might guess that the
following relationships hold:

- Output = 2*Input (
*i.e.*,* y* = 2*x*)
- Output = Input + 2 (
*i.e.*,* y* = *x *+ 2)
- Output = 6 – Input (
*i.e.*,* y* = 6 – *x*)
- Output = 0.5*Input + 3
*(i.e.*, *y* = 0.5*x* + 3)

By dragging the bottom point, students can test their
conjectures against new input-output pairs. The image in **figure 5** was generated by dragging the “input” point 4 units to the
left.

**Fig.
5** This graph results when the point along
the input number line is dragged 4 units to the left.

Of the original conjectures, only the last one holds for
both input values. Clicking on the “Show Equation” checkbox, students see that
the relationship between input and output is, indeed, defined as

*y* = 0.5

*x* + 3. As

**figure 6**
suggests, teachers (and students) can type in new functions into the input box
and explore new rules.

**Fig.
6 **Students can reveal

** **the relationship between input and output by clicking on the “Show
Equation” checkbox.

Instructionally, the approach is similar to the “Guess My
Rule” game with several important differences:

- The dynagraph emphasizes the
*continuous change *of both the input and output values of the
function.
- The interactive sketch gives students and teachers a way
to
*manipulate inputs directly* through
dragging, encouraging students to see variables as *quantities that vary*.

Dynagraphs were first envisioned by Goldenberg, Lewis, and
O’Keefe (1992) as a means to bridge discrete, numerical representations of
function (e.g., tables of values, function machines) with more sophisticated,
abstract representations that students encounter in the later grades (e.g.,
formulas and equations, Cartesian plots).

**From
Dynagraphs to Cartesian Plots** The output number line of a dynagraph can be rotated 90
degrees to yield a dynagraph that resembles the Cartesian plane (as shown in

**fig. 7**). Constructing lines through
“input” and “output” points perpendicular to the number lines, the

*x-y* pairs are plotted by tracing the
intersection of the perpendiculars (as shown in

**fig. 8**).

**Fig.
7** This dynagraph image, showing an output
number line rotated 90 degrees, is available at

http://bit.ly/dynagraph2.

**Fig.
8** This Cartesian plot is available at

http://bit.ly/dynagraph3.

Dynagraphs are a powerful representation
to help students better grasp the idea of function. Traditionally, students in
the upper middle grades explore graphs of functions wholly within the Cartesian
plane. Too often, this approach doesn’t adequately connect back to experiences that
students have had with functions in the earlier grades. By giving students a
way to explore functions in one dimension, dynagraphs help students focus on
relationships between inputs and outputs while considering functions as
objects. Dynagraphs may also be used to help students transition to Cartesian
graphing.

**References** Goldenberg,
P., P. Lewis, and J. O’Keefe. 1992. “Dynamic Representation and the Development
of a Process Understanding of Function.” In

*The
Concept of Function: Aspects of Epistemology and Pedagogy, *edited by Ed
Dubinsky and Guershon Harel, pp. 235–60.

*MAA
Notes no. 25*. Washington, DC: Mathematical Association of America.

Huinker,
D. 2002. “Calculators as Learning Tools for Young Children’s Explorations of Number.”

*Teaching Children Mathematics *8 (February):
316–21.

King, S. 2002. “Sharing Teaching Ideas:
Function Notation.”

*Mathematics Teacher
95 *(April): 636–39.

Reeves, C. 2005/2006. “Putting Fun
into Functions.”

*Teaching Children
Mathematics *12 (December 2005/January 2006): 250–59.

Steketee,
S., and D. Scher. 2011. “A Geometric Path to the Concept of Function.”

*Mathematics Teaching in the Middle School *17
(August): 48–55.

Michael
Todd Edwards is an associate professor of mathematics education at Miami
University in Oxford, Ohio. He is the coeditor of

*Contemporary Issues in Technology and Mathematics Teacher Education*,
executive editor of the

*North American
GeoGebra Journal*, and codirector of the GeoGebra Institute of Ohio. His
research interests focus on the teaching and learning of school mathematics
with technology (specifically, dynamic mathematics software), ethical issues
surrounding the use of free software and the free software movement, and
writing as a vehicle to learn mathematics at all levels of instruction.

Jennifer Nickell is working toward her
PhD in mathematics education at North Carolina State University in Raleigh. Her
research interests focus on preparing teachers to teach statistics and
effectively incorporate technology into the classroom.