Student Inquiry and the Rascal Triangle

Abstract

Those of us who teach Mathematics for Liberal Arts (MLA) courses often underestimate the mathematical abilities of the students enrolled in our courses. Despite the fact that many of these students suffer from math anxiety and will admit to hating mathematics, when we give them space to explore mathematics and bring their existing knowledge to the problem, they can make some amazing mathematical discoveries. Inquiry-based learning (IBL) is perfect structure to provide these type of opportunities. In this paper, we will examine one inquiry-based investigation in which MLA students were given the space to look for patterns which resulted in some original discoveries.

Generalized Rascal Triangles

“…In the Fall 2015, I challenged students in one of my Mathematics for Liberal Arts classes to find patterns in the Rascal Triangle. To my delight, they discovered several properties that, as far as I can tell, were unknown at the time; see Student Inquiry and the Rascal Triangle [3] for details about my student's work. Further investigations showed that these properties were also present in Generalized Rascal Triangles; we conclude by presenting proofs of several of these properties, as well as some others, for Generalized Rascal Triangles.…”

“…The fact that both Equations 1 and 2 can be used to generate the Rascal Triangle was intriguing to me and I assumed that the Rascal Triangle was uniquely defined by either one of the two equations; so I began trying to prove that Equation 2 implied Equation 1 or vice versa. In addition, I followed Julian Fleron's lead and had some of my mathematics for liberal arts classes look for patterns in the Rascal Triangle, and to my delight they also made some original discoveries [3]. During the following summer, while exploring the patterns found by my students, I realized that there were other number triangles for which one equation held for the interior entries but the other did not, as will be shown below.…”