### Electronic Proceedings of the Eleventh Annual Conference on
Technology in Collegiate Mathematics

*CONTRIBUTED PAPER: 11-C44*
### Product of Shears

Gina M. Foletta

Northern Kentucky University

Department of Mathematics

Highland Heights, KY 41099-1700

Phone: (606) 572-6349

E-mail: foletta@nku.edu

#### ABSTRACT

In our college geometry course, shears are a small component of our treatment of
transformational geometry. Yet, shears are important because they counteract the
commonly held belief that all transformations are isometries - or at least similitudes.
While I was consulting for the CAS-Intensive Mathematics Project (A curriculum
development project [NSF-ESI-9618029] directed jointly from The Pennsylvania State
University and The University of Iowa.) this summer, a colleague posed a question:
"Are you aware that rotations are implemented on some dynamic geometry tools as a
product of three shears?" After my initial response of skepticism, I began to explore
the question.
My investigation resulted in the following theorem: A rotation about the origin
is the product of three shears. In the proof I used shears about the x-axis and
y-axis. The paper elaborates on this theorem and discusses several teaching issues.