A Geometric Approach to Functions

by Karen Hollebrands, Allison McCulloch, Daniel Scher, and Scott Steketee

Fostering an understanding and appreciation of the deep, beautiful threads that unite seemingly disparate areas of mathematics is among the most valuable outcomes of teaching. Two such areas that are ripe for bridge building—functions and geometric transformations—are the focus of our NSF project, Forging Connections Through the Geometry of Functions. In this post, we describe the pedagogical benefits of introducing students to functions through the lens of geometric transformations.

Geometric Transformations as Functions

The most common representations of functions are symbolic and numeric in nature. This emphasis on number limits students’ images of the variety of mathematical relationships that can be represented as functions. As such, it contributes to common student misconceptions. Students may conclude that:

• every function turns an input number into an output number;
• every function can be expressed as an algebraic formula;
• a formula is the primary representation of a function, and all other representations derive from it; and
• the ultimate test of a function requires graphing it in rectangular coordinates and applying the vertical line test.

Although students investigate reflections, translations, rotations, dilations, and glide reflections in a geometry course, they typically do not regard them as functions; the functions they encounter in algebra always have numbers as input and output. We can expand students’ horizons and deepen their concept of function by treating geometric transformations as functions that take a Euclidean point as input and produce another point as output. Coxford and Usiskin pioneered this approach a half century ago in their ground-breaking Geometry: A Transformation Approach, but very few of today’s geometry students encounter it.

In Figure 1, a student has used three Web Sketchpad tools to construct the independent variable x, the mirror m, and the reflected dependent variable r_m(x). The student then drags point x and observes the traces of both point x and r_m(x). This sensorimotor experience introduces students to four mathematical ideas.

1. Function notation is meaningful. The use of function notation gives students language to describe the specific elements that constitute the function: independent variable x, function rule r_m (“reflect in m”), and dependent variable r_m(x) (“the reflection in m of x”).
2. Functions need not be algebraic formulas with numeric inputs and outputs. Point r_m(x) depends on x: Students can drag x in order to make point r_m(x) move, but cannot drag r_m(x) by itself.
3. Variables really vary. As students drag independent variable x, red and blue traces memorialize the kinesthetic experience of varying the variable. The traces form a pictorial record of the dynamic interaction and help students analyze the covariation.
4. Relative rate of change can be observed and described. By dragging x, students observe that x and r_m(x) always move at the same speed, but not always in the same direction, and they can investigate how to drag x so the variables move in the same direction or in opposite directions.

Figure 1

Constructing a Dynagraph

Having explored reflection and other geometric transformations in two-dimensional Flatland, students then restrict the domain of these transformations into the Lineland (one-dimensional) environment of a number line (Abbott, 1886). They focus in particular on connecting the geometric behavior of dilation and translation to the observed numeric values of their variables on a number line.

In Figure 2, students use the Number Line, Point, and Dilate tools to create a point restricted to the number line and dilate it about the origin to obtain a point labeled D_0,s(x). While this notation may at first seem daunting, it actually may be less mysterious than the traditional f(x) language. Rather than write out or speak all the words “the Dilation (of x) about center point 0 by scale factor s,” we abbreviate the important parts with just single letters to encapsulate this long-winded phrase. The notation f(x) does that, to an extent, too, but leaves f a mystery that is unraveled somewhere else (on the right hand side of the = in the definition statement) but never seen again. Algebraic notation is an “abbreviation” of what can be said in English (and used to be), but much more clumsily.

Students measure the coordinates of x and D_0,s(x) and drag x to compare the values. When asked to describe how D_0,s(x) moves when x is dragged, a student might respond, “As I drag x, D_0,s(x) moves faster. It seems to move twice as fast, and I notice its value is always twice the value of x. I wonder if its speed is related to the scale factor s.” By experimenting with different scale factors, the student concludes that s represents the relative speed of D_0,s(x) with respect to x, and that the coordinates produced by this dilation satisfy D_0,s(x) = x·s. Students can then experiment with a translation restricted to the number line and conclude that this new function, translation by a vector parallel to the number line and of directed length v, causes the two variables to move at the same speed, and satisfies the equation T_v(x) = x + v. This understanding—that multiplication and addition of numbers are the equivalents of dilation and translation of points—represents a deep connection between algebra and geometry.

Figure 2

Students are now ready for a new task: What happens when you dilate x and then translate the dilated image; in other words, how does the composite function T_v(D_0,s(x)) behave? Students’ first attempts at this task becomes visually confusing with three variables and a vector stumbling over each other on the same number line. To alleviate the confusion, we introduce a Transfer tool that moves the dependent variable to a parallel number line, separate from but aligned with the first. In Figure 3, students use this tool to construct a second number line parallel to the original. This visual representation of a function, with the independent variable x on one number line and the dependent variable T_v(D_0,s(x)) on a parallel number line is known as a dynagraph (Goldenberg, Lewis, & O’Keefe, 1992).

Figure 3

Students who construct T_v(D_0,s(x)) = x·s + v have built a linear function. This may seem a lot of effort to get to what is more typically presented on a Cartesian graph with sliders that control the values of s and v (generally labeled mysteriously as m and b), but the geometric approach has advantages. Note that the terms slope and y-intercept reference the Cartesian-graph representation, without directly communicating the fundamental underlying concepts of relative rate of change and starting value. Lost in this representation is an understanding of the choreographed dance between independent variable x and dependent variable T_v(D_0,s(x)). Though this information is encoded in the Cartesian graph and its accompanying algebraic form, it’s difficult for many students to tease the deeper concepts out of these representations. A dynagraph makes this behavior more accessible by allowing students to drag point x and compare its speed and direction with the behavior of T_v(D_0,s(x)) on a parallel number line. To answer the questions below, all of which focus on motion, students experiment with the values of s and v to create linear functions that match the desired behaviors.

• As x is dragged, T_v(D_0,s(x)) moves in the same direction as x, but 3 times as fast as x.
• As x is dragged, T_v(D_0,s(x)) is stuck at 2.
• As x is dragged, T_v(D_0,s(x)) moves at the same speed as x, but in the opposite direction.
• As x is dragged, T_v(D_0,s(x)) moves in the opposite direction as x, but twice as fast.
• As x is dragged, T_v(D_0,s(x)) moves 2 times as fast as x, and is always 4 behind D_0,s(x).

The Cartesian Connection

To conclude, students create the Cartesian graph of a linear function using geometric transformations. As Figure 4 illustrates, students start with the same initial tools that they used to create a dynagraph, but this activity’s Transfer tool rotates a variable by 90°, transferring it to a vertical number line perpendicular to the original, horizontal number line. After using this tool to rotate D_0,s(x) to a vertical axis and translating by vector v, students use the Perpendicular tool to construct lines that keep track of the horizontal location of x and the vertical location of T_v(D_0,s(x)). They then construct a traced point at the intersection of these horizontal and vertical lines and drag x to see how the traced point’s motion corresponds to the behavior of the two variables. After performing the construction, students try different values for the scale factor s and the translation vector v, and they observe not only how changing the scale factor affects the speed of T_v(D_0,s(x)) relative to x but also how the speed of the dependent variable determines the steepness of the traced line. After making this observation, it’s relatively easy for students to propose that steepness could be measured numerically based on how far T(D(x)) moves relative to the movement of x, leading to a mathematical definition of slope as Δy/Δx.

Figure 4

Conclusion

By using web-based dynamic mathematics software and tools tailored to carefully structured tasks, students can enact geometric transformations as functions, create them, manipulate them, and experiment with them. In the course of their explorations they can develop a solid understanding of geometric transformations, explore connections between geometry and algebra, and construct and shed light on linear functions by using a dynagraph representation.

By beginning with R²→R² functions (transformations in the Euclidean plane) and connecting them to R→R functions in algebra, these activities can help prepare students for later study of complex (C→C) functions, and functions with three-dimensional domains and ranges (R³→R³). A further benefit is the gentle visual introduction of the concept and notation of function composition.

Pedagogically, the constructive nature of activities such as these has the potential to engage students, to give them opportunities to assess their own work, to encourage mathematical discussions, and to help students bridge the gap between the concrete, physical world and the profound elegance of abstract mathematical insights.

Acknowledgments

This post is based in part upon work supported by the National Science Foundation under NCSU IUSE award 1712280 (July 2017 through June 2019) and KCP Technologies DRK-12 award ID 0918733 (September 2009 through August 2013). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

We are deeply grateful to McGraw-Hill Education for making Web Sketchpad available for the activities described in this post.

References

Abbott, E. A. (1884). Flatland: A romance of many dimensions. San Antonio, TX: Eldritch Press.

Coxford, A. F., & Usiskin, Z. (1971). Geometry: A transformation approach. Laidlaw Brothers Publishers.

Goldenberg, P., Lewis, P., & O’Keefe, J. (1992). Dynamic representation and the development of a process understanding of function. In E. Dubinsky & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 235-260). Washington, DC: Mathematical Association of America.