1. 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),
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),