Ben Bazak Harold Mick
Patrick Henry High School Retired Faculty, Virginia Tech
We present an “alternative” approach for applying transformations to problems of the type: (1) given a graph, write its equation, and (2) given an equation, sketch its graph. Our approach relies heavily on coordinates of points in the coordinate plane, where a point is an ordered pair (x, y) of numbers. Equations describe the relationship between coordinates for exactly those points on its graph; and graphs are geometric pictures made up of exactly those points whose coordinates are in the relationship described by the graph’s equation. We construct sequences of transformations to move points from one graph to another, which build up algebraic expressions for individual coordinates. The movement of transformations is initially from unknown graphs to known graphs. We create undoing (inverse) transformations for movement in the other direction — from known graphs to unknown graphs, which then become known.
We have presented details of our approach in articles and conference papers: We gave session participants a 10-page handout at the VCTM 2017 Annual Conference (2017); we gave session participants a 14-page handout at the VCTM 2018 Annual Conference (2018); and we have two publications in the Virginia Mathematics Teacher Journal (2017, 2018). We call our approach an alternative approach because it is decidedly different from current practices.
Current practices for dealing with transformations are determined by secondary school textbooks and, in the Commonwealth of Virginia, by the Mathematics Standards of Learning (SOLs). One current emphasis is on substitution (replacement) rules. For instance, Murdock, J., Kamischke, Ellen., & Kamischke, Eric (2010) demonstrate the use of substitution rules in showing teachers how to help students determine the order of applying transformations:
Point out that there is a specific order for any given set of transformations. Suppose students are asked to describe the two transformations that changed f(x) = x2 into g(x) = [(x + 2)/3]2. They should start by looking at how the parent function, f(x), has been modified. In function form, this equation would look like g(x) = f[(x + 2)/3]. First, x has been replaced by x/3, representing a horizontal dilation by a factor of 3. Next, x has been replaced by x + 2, meaning a horizontal shift 2 units to the left. In this situation, f(x) was dilated horizontally by a factor of 3 and then shifted horizontally −2 units; each new y-value is the result of multiplying an x‑value by 3 and then subtracting 2. Note that the order of substitution in not reversible and that this does not follow the logic of the order of operations. If you first replace x with x + 2 and then replace x with x/3, you get g(x) = f(x/3 + 2). [Big Idea] When a graph is transformed, each variable in the graph’s equation is replaced with a variable and a constant representing that aspect of the transformation. (p. 230)
In addition to substitution rules, current practices rely on equation forms that show the sequence of transformations embedded in equations. For an example of using equation forms, here is an excerpt taken from Virginia’s Trigonometry SOL (2016):
Standard form of the trigonometric functions may be written in multiple ways (e.g., y = Asin(Bx + C) + D or y = Asin[B(x + C)] + D.
Describe the effect of changing A, B, C, or D in the standard form of a trigonometric equation.
Sketch the graph of a function written in standard form by using transformations for at least a two-period interval, including both positive and negative values for the domain.
What order of applying transformations does the standard form, y = Asin(Bx + C) + D, emphasize? Does the mnemonic CBAD (pronounced C’BAD) apply? Will the order of application change if the direction of movement changes between the graph of the standard form and the parent? How do the forms y = Asin[B(x + C)] + D and y = Asin(Bx + C) + D differ regarding order of application? Is the distinction between expressions B(x + C) and Bx + C a clue?
Upon reflection, it appears that the excerpt from the textbook, which deals with the distinction between the order of substitutions (x + 2)/3 and x/3 + 2 is addressing the same order‑of‑application complexity as the trigonometry SOL in distinguishing between B(x + C) and Bx + C. In short, substitution rules and standard forms are driving the application of transformations with current classroom practices.
The National Council of Teaches of Mathematics (NCTM) has published Catalyzing Change in High School Mathematics: Initiating Critical Conversations (2018) and made the book available to the public at their 2018 Annual Meeting in Washington DC this Spring. NCTM wants to change the high school curricula to make it more relevant for today’s world. In Catalyzing Change, NCTM is specific with the mathematics topics, and puts considerable emphasis on transformations throughout their discussion. Will the secondary school mathematics community embrace Catalyzing Change? Is the secondary school mathematics community okay with current practices regarding transformations?
We offer our alternative approach to transformations as a catalyst for initiating a critical conversation about current practices and future directions on this topic.
Bazak, B., & Mick, H. (2018). The shape of ordered pairs: Connecting graphing to big ideas. Paper presented at the VCTM 2018 annual conference, Moving mountains with mathematics. Radford, VA. Session paper retrieved from vctm.org/resources/Conference and PD/2018 Conference/2018_Sessions/Session 62.pdf.
Bazak, B., & Mick, H. (2017). Graphing in the future: A unifying approach. Paper presented at the VCTM 2017 annual conference, A community of math heroes: Educate, encourage, inspire. Harrisonburg, VA. Session paper retrieved from vctm.org/resources/2017 Conference/Conference Resources/Session 003.pdf.
Mick, H., & Bazak, B. (2018). The shape of ordered pairs: Last step. Virginia Mathematics Teacher, 44(2). Retrieved from vctm.org/The-Shape-of-Ordered Pairs- Last-Step.
Mick, H., & Bazak, B. (2017). The shape of ordered pairs. Virginia Mathematics Teacher, 44(1), 44-50.
Murdock, J., Kamischke, Ellen., & Kamischke, Eric. (2010). Discovering advanced algebra: An investigative approach [Teachers second edition], see Chapter 4, p. 230. Emeryville, CA: Key Curriculum Press.
National Council of Teachers of Mathematics (NCTM). (2018). Catalyzing change in high school mathematics: Initiating critical conversations. Reston, VA: NCTM, Inc.
Virginia Department of Education. (2016). Trigonometry 2016 standards of learning. [Web post]. Retrieved from www.doe.virginia.gov/testing/sol/standards_docs/mathematics/2016/cf/trigonometry‑cd.pdf.