The mathematical concept of slope can be made real through a set of simple, inexpensive, and safe experiments that can be conducted in the classroom or at home. The experiments help connect the idea of slope with physical phenomena related to surface tension. In the experiments, changes in surface tension across the surface of the water, which correspond to greater slopes on the graph, lead to increased motion of the fluid. The mathematical content, targeted to middle school and high school students, can be used in a classroom or workshop setting and can be tailored to a single session of thirty to ninety minutes.
When understood and applied appropriately, mathematics is both beautiful and powerful. As a result, students are sometimes tempted to extend that power beyond appropriate limits. In teaching statistics at both the high school and college level, I have found that one of students' biggest struggles is applying their understanding of probability to make appropriate inferences.