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Exploring Linear Functions using Arrow Diagrams Melva Grant

mathematical experience that is accessible to learn ers with diverse mathematical backgrounds and un derstandings. An arrow diagram is a graphical rep resentation for ordered pairs ( x , y ) of inputs and outputs. To create an arrow diagram, draw two ver tically oriented parallel axes with equal scales; the left axis is used to plot the input values (i.e., inde pendent, x ), the right axis is used to plot the output values (i.e., dependent, y ), and rays or lines are used to connect input values to output values. I en courage learners to align their scales (i.e., the 0’ s for each axis are at the same height) to enable com parisons between graphs. Examples of arrow dia grams are provided showing a) several ordered pairs; and b) the equation, y = x (see Figure 1). For more information about arrow diagrams, explore the Underground Mathematics website sponsored by the University of Cambridge (n.d.) that provides an introduction and a mapping diagram activity to explore linear functions of the form f ( x ) = ax + b (for all real numbers). This website offers a more expansive activity than can be described in this pa per due to time and platform; the online version in cludes an interactive GeoGebra construction tool for exploration. This inquiry, like the Underground Mathematics inquiry, is about relationships between linear func tions and their arrow diagram representations. Lin ear functions and their graphs within a Cartesian

Introduction

While working with in - service elementary and middle school mathematics teachers and coaches enrolled in a Functions and Algebra course within our university ’ s mathematics specialists ’ program, we engaged in an exploration that encourages com munication, uses multiple representations, and of ten leads to interesting mathematical connections beyond the conceptual ideas related to linear equa tions and their graphical representations. A chal lenge when working with in - service teachers is to find mathematics tasks that are appropriate or near appropriate for the grade levels they teach, yet is sufficiently rigorous to engage teachers as learners who have diverse mathematics understandings (Hsu et al., 2012). These researchers recommended creating equitable learning environments that allow all learners to develop new understanding by con tributing to the learning process. This is the chal lenge that exists when working with mathematics teachers and teacher leaders, and is the impetus for presenting this rich task. This inquiry - oriented ac tivity was designed for practicing mathematics teachers enrolled in a mathematics specialist pro gram. The inquiry uses a graphical representation that I call arrow diagrams, but they are also referred to as mapping diagrams. The arrow diagrams afford a

Virginia Mathematics Teacher vol. 47, no. 1

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