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coordinate system are a regular part of the second ary mathematics curricula . This inquiry, while out side the regular curricula, offers a novel opportuni ty for learners to reflect on and construct new mathematical understandings that are connected to extant mathematical knowledge. The focus for in quiry is to find patterns among graphical represen tations of linear functions represented using arrow diagrams (Baroody & Coslick, 1998). My experi ence working through this inquiry with in - service teachers has consistently led to reflective, coherent, and connected mathematical learning and discus sions. The types of concepts and the mathematical engagement that emerge during this inquiry are documented in the Algebra standard and the mathe matical practices of the Common Core Standards— representing linear functions graphically and look ing for structure and regularity (CCSSM, 2010). Additionally, researchers and others have suggest ed increasing opportunities for learners to experi ence and participate in mathematical reasoning and problem solving as a key element for 21 st Century living and decision making (e.g. Strutchens & Quander, 2011). Thus, confirming the value of this inquiry for expanding mathematics learning and understanding for all learners, including in - service mathematics teachers. To prepare for this inquiry in my courses, learners need an introduction to arrow diagrams because they are not typically included in secondary school mathematics curricula. For my classes, the learners prepare for the exploration by researching arrow diagrams on the Internet and then independently creating arrow diagrams given several ordered pairs and a quadratic equation (i.e., y = x 2 + 1) prior the class for this inquiry. The first day of the in quiry begins with learners sharing their independ

Figure 1: Arrow diagram examples for introduction

ent explorations about arrow diagrams. Invariably, someone introduces a diagram similar to that shown in Figure 1 that shows the arrow diagram for the ordered pairs assigned. In class, I used a document camera for teachers to share their draw ings and to explain what they did and why. Before moving on, I ensure that a good example of an ar row diagram was displayed. In addition, a brief dis cussion about the arrow diagram for y = x can ac complish this (see Figure 1b). I clarify the limita tions for the graphical representations that we will produce during the inquiry – the x - axis on the left, the y - axis on the right, using equal scales for both axes, and 0’ s aligned. If learners notice that the rays are parallel for the y = x arrow diagram, make note of their interesting observation and suggest re visiting it later after more arrow diagrams have been created. Before creating arrow diagrams, learners share their knowledge about relationships between linear functions and their graphs in the Cartesian coordi nate plane. In the mathematics specialists ’ course, mathematics teachers came up with an expected

Table 1: A variety of linear functions by type

Virginia Mathematics Teacher vol. 47, no. 1

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