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linear equations with no slopes, such as x = a, the rays will intersect on the x - axis. Reflecting on the arrow diagrams shown in Figure 3, possible conjec tures include: for functions with slopes, - 1 < m < 0, the rays intersect between the axes, and their inter sections are closer to the y - axis than the x - axis; and the smaller the slope the closer the rays ’ intersec tions are to the y - axis. Considering Figure 2, a rea sonable conjecture is a linear function with slope, m = - 1, the rays intersect at the midpoint between the axes. Further examination of Figure 3 shows the influence of the y - intercept, b. Conjectures about b include: the function ’ s y - intercept influ ences the height of the intersection—given two functions, one with b = - 4 and the other with b = 6, the intersecting rays will be close to - 4 and 6, re spectively; and the function with the larger y intercept will have the higher point of intersecting rays. Finally, examining Figure 4, a conjecture is: for functions with slopes, m > 1, the rays intersect left of the x axis. For interested readers, create more arrow diagrams to determine additional con jectures. Using the arrow diagrams created in one course, mathematics teachers came up with the following conjectures, which they developed arguments to support or refute using the set of arrow diagrams created during the class. In several instances, they had to convince their peers of the validity of their conjectures and in other instances they proved con jectures invalid. For this paper, I made no effort to edit their phrasing and during the course I did not validate their mathematical thinking. My role was to ask questions to probe their thinking to improve clarity and to seek connections. Consider their con jectures in light of what you have learned from your reading and the shared arrow diagrams: Which conjectures, if any, can you support as val id? Which, if any, can you refute as invalid?

• Negative slopes, the arrows crisscross • When the slope is 1, the lines are parallel • When the slope is zero, the arrows point of con vergence is on the y - axis • When the slope is greater than 1 the arrows point of convergence is left of the x - axis • When the slope is positive the lines do not in tersect between the x and y axes • The bigger the value of the y - intercept, the in tersection is closer to the x - axis The mathematics teacher learners discussed signifi cantly more than what was recorded during the course. For example, several of the learners ex plored and found what they believed to be the point of intersection beyond the y - axis. Others consid ered using arrow diagrams as a way to connect lin ear equations to an introduction of Geometry top ics, such as symmetry of triangles formed by the lines formed by the rays and axes and identifying instances when these triangles are congruent. Clearly, some conversations included mathematical connections beyond the anticipated exploration of patterns formed from linear functions using arrow diagrams. The vast array of mathematics concepts and the substantive nature of the discourse among a group of diverse adult learners offer compelling ev idence that this task is indeed a rich task worth a try with mathematics teachers, and perhaps even secondary students. Baroody, A. & Coslick, R. (1998). Fostering Chil dren's Mathematical Power: An Investiga tive Approach to K - 8 Mathematics Instruc tion . Mahwah, New Jersey: Lawrence Erl baum Associates. Common Core State Standards for Mathematics . (2011). Common Core State Standards Ini tiative 2010. Retrieved May 10, 2011, from http://www.corestandards.org/the standards/mathematics/introduction/ standards - for - mathematical - practice/. Hsu, E., Kysh, J., Resek, D., & Ramage, K. (2012). Changing Teachers' Conception of Mathe matics. NCSM Journal of Mathematics Ed- References

Teachers ’ conjectures based on patterns seen in ar row diagrams:

• Positive fractional slopes, the arrows converge • Positive integer slopes greater than 1, the ar rows diverge

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