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Teaching Students about Functions with Dynagraphs

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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.

Figure 1: Students record their data 

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.

Figure 2: Illustration of a third-grade version of a function machine task 

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.

Figure 3: Graphical, tabular, and symbolic representations of  f(x) = 2x + 1  

Fig. 3 Graphical, tabular, and symbolic representations of are available at 

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.

Figure 4: Dynagraph sketch
Fig. 4 This dynagraph example is available at  

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

Figure 5: Resulting graph
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 = 0.5x + 3. As figure 6 suggests, teachers (and students) can type in new functions into the input box and explore new rules.
Figure 6: Graph showing results of clicking Show Equation checkbox
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).
Figure 7: A dynagraph that resembles the Cartesian plane 
Fig. 7 This dynagraph image, showing an output number line rotated 90 degrees, is available at
Figure 8: Cartesian plot
Fig. 8 This Cartesian plot is available at
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.
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 EdwardsMichael 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 NickellJennifer 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. 

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