Abstract. In this paper we consider a refinement, due to Nathanson, of the Calkin-Wilf tree. In particular, we study the properties of such trees associated with the matrices Lu = 1 0 u 1 and Rv = 1 v 0 1 , where u and v are nonnegative integers. We extend several known results of the original Calkin-Wilf tree, including the symmetry, numeratordenominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin-Wilf tree.

For positive integers u and v, let Lu = 1 0 u 1 and Rv = 1 v 0 1 . Let Su,v be the monoid generated by Lu and Rv, and Gu,v be the group generated by Lu and Rv. In this paper we expand on a characterization of matrices M = a b c d in S k,k and G k,k when k ≥ 2 given by Esbelin and Gutan to Su,v when u, v ≥ 2 and Gu,v when u, v ≥ 3. We give a simple algorithmic way of determining if M is in Gu,v using a recursive function and the short continued fraction representation of b/d.

Students graduating from high school in the United States are often underprepared, unaware of, and surprised by the rigors of college courses, especially mathematics, and consequently stumble in their first year. A pilot intervention, the Academic Inventory Module (AIM) for Success in Mathematics project, at a minority-serving institution was offered to incoming freshmen students, which included a nine-hour mathematics preparation course and participation in a peer-led workshop in their first mathematics credit-bearing course. The goal of the AIM project was to establish a seamless pathway from high school to college learning of mathematics through the Peer-Led Team Learning model of instruction. The principal activities of the project were to offer a bridge program preparing students for a combined algebra and geometry course and to create a peer-led community to support freshmen enrolled in the course. Results from this study showed students participating in peer-led workshops had significantly higher grades and lower failure and withdrawal rates than those who did not participate.

The set of positive linear fractional transformations (PLFTs) is partitioned into an infinite forest of PLFT Calkin-Wilf-trees. The roots of these trees are called orphans. In this paper, we provide a combinatorial formula for the number of orphan PLFTs with fixed determinant D. Then we provide a way of determining the orphan of a PLFT Calkin-Wilf-tree for a given PLFT. In addition, we show that every positive complex number is the descendant of a complex (u, v)orphan. Date