A Spatial-Visual Approach to Optimization and Rates of Change Robyn Ruttenberg-Rozen Ami Mamolo Walter Whiteley York University UOIT York University 1 Problem 5 Key learning And Big Ideas . Given a square sheet of material, cut equal squares from the corners and fold up the sides to make an open-top box. How large should the square cut-outs be to make the box contain maximum volume? A B Some big ideas that resulted from exploration of our network of tasks: * The volume of the boxes can change * Boxes with different shapes could have the same volume. * There is a largest volume. *The maximum volume is not: at either extreme, in the cube shape, or the ‘middle’ between extremes. *Volume and surface area can be physically represented, and physically compared in multiple consistent ways. *Change in volume between pairs of boxes involves both volume lost and volume gained as the cut size is increased. *Given the uniform thickness of these gains and losses (the size of the increase in the cut), these changes in volume (loss and gain) between pairs of boxes can be compared with clarity by naïve overlay strategies 2 Optimization and Rate of Change Conceptual understanding of optimization and rate of change can be challenging for students. Spatial-visual tools increase student understanding of conceptual ideas in calculus (Berry and Nyman 2003). Considering the difficulties of underlying conceptual understandings that key concepts in calculus pose for learners in higher education, researchers have recommended building the underlying conceptual understandings of calculus, including rate of change and optimisation, as early as elementary school. Network of Tasks to Support Learning 4 Scaffolding The culminating task (first developed in Whiteley, 2007): A B We introduced tasks to support the learning, each differentiated to the differing needs of the students i.e.: Graphical Representations: Through Geometer’s Sketchpad, highlights important connections between average rates of change and slopes of secant lines, and instantaneous rates of change and slope of tangent lines Filling Task: Introduces volume comparison that does not require calculating one volume first and then the other An example of an implementation can be found in Whiteley & Mamolo (2011) 3 In order to enable everyone access to a conceptual understanding of optimization and rate of change, we scaffolded vertically across grade levels, elementary, high school and university, and horizontally across ability levels in the same classroom. We used the tools of scaffolding developed by Anghileri (2006). Through and within our network of tasks we provided multiple entry points for the students. The passage through the tasks were not sequence dependent, a class of children or a child experienced different trajectories depending on their learning needs. an example of a scaffolded task network * References 1. Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning. Journal of Mathematics Teacher Education, 9(1), 33-52 2. Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 479-495 3. Whiteley, W. (2007) Visual reasoning: rates of change without derivatives OAME Conference Georgian College, Barrie, Ontario 4. Whiteley, W., & Mamolo, A. (2011). The Popcorn Box Activity and Reasoning about Optimization. Mathematics Teacher, 105(6), 420-426.