**Mathematics Abstracts**

EXTREME VALUES OF FUNCTIONS OF SEVERAL VARIABLES

**Derrick Bennett**

*Faculty Adviser: Hansun To, Ph.D.*

The abstract of this presentation is on solving for minimum values of multivariable functions using Lagrange’s Theorem. The problem was proposed in the October 2008 issue of Mathematics Magazine. The problem stated let x, y, and z be positive real numbers with x+y+z = xyz. Find the minimum value of √1+x^2 + √1+y^2 + √1+z^2. The solution of the stated problem was found to be x=y=z=√3, which gave a minimum value of 6.

PROBLEM SOLVING WITH MODEL DRAWING

**Richard Bisk, Ph.D.**

The model drawing approach to problem solving is used extensively in the math curriculum of Singapore. It takes students from the concrete to the abstract via an intermediary pictorial stage. Students create bars and break them down into “units.” The units create a bridge to the concept of an “unknown” quantity that must be found. Students can learn to use this strategy in the primary grades and continue with it through the middle grades. At Worcester State College, we teach this technique to our pre-service elementary teachers.

MATHEMATICS PREPARATION AND ACHIEVEMENTS AND THE COLLEGE EXPERIENCE

**Richard Bisk, Ph.D., and Mary Fowler, Ph.D.**

Mathematics is an integral part of the educational experience of all students. Students arrive at Worcester State College with varying levels of mathematical preparation and are given tests so they are placed into a class for which they are well prepared. This poster presents results of investigations studying the mathematical experience of WSC students. How does a student’s level of mathematical preparation affect his or her success in college? How does a student’s mathematical achievement while at WSC relate to his or her likelihood of college completion?

FLIGHTS FROM ORLANDO: IS THERE A CONNECTION BETWEEN DISTANCE AND PRICE?

**Thomas Brennan and Abigail Chmielecki**

*Faculty Adviser: Maria Fung, Ph.D.*

We analyze data from two different variables, study their correlation, examine outliers, and analyze the validity of the linear model.

EVALUATING A FRACTION OF THE FORM Z/(Y+Z)

**Meghan Coyne**

*Faculty Adviser: Hansun To, Ph.D.*

The objective of this problem is to find z/(y + z), given that z/(x + y)= a and y/(x + z)= b. There are two cases for solving this problem: y = z and y ≠ z. In the trivial case, y = z, it is easy to show z/(y + z)=1/2. For the second case, y ≠ z, with algebraic transformations on z/(y + z), and by recognizing algebraic similarities between a + 1 and b + 1 it can be shown that z/(y + z) = (ab + a)/(a + 2ab + b).

FINDING WEINER INDEX FOR THE GRID AND COMB GRAPHS ON 2N VERTICES

**Joe Fredette**

*Faculty Adviser: Hansun To, Ph.D.*

In Graph Theory, we can define the “Weiner Index,” which is the sum of all the shortest paths in a given graph. Given two specific indexed sets of graphs, namely the Grid and Comb graphs on 2n vertices, the aim of this paper was to characterize the Weiner Index for all the Grid and Comb graphs. In this paper, we do this by using techniques from signal analysis and from the theory of recurrence relations.

MOIRÉ FRINGES

**Joe Fredette**

*Faculty Adviser: Maria G. Fung, Ph.D.*

Moiré Fringes, which are most often noticed as interference patterns on a TV screen or through a window screen, are the small bands caused by the interference of two iterated patterns. We can describe this interference as a function of the gradients of two iterated functions. This gradient does not only predict the existence of Moiré Fringes, but also their periodicity and density. In this project, we describe how this metric for Moiré fringes works, and provide some examples including mathematical code, which generates images, and associated informational graphs, which show Moiré Fringes. Math majors Jonathan Lussier and Richard Ouellette collaborated on this project.

CHALLENGING PROBABILITY PROBLEMS

**Maria G. Fung, Ph.D.**

This is a conference presentation given at the Northwest Mathematics Conference for middle and high school teachers. We consider the Sock Drawer and the Chuck-a-Luck classic probability problems. We illustrate ways to turn them into successful activities for the middle and high school classroom.

MATH FACULTY AS PARTNERS IN TEACHING NON-EUCLIDEAN GEOMETRY COURSE FOR K-12 TEACHERS

**Maria G. Fung, Ph.D.**

We discuss how the Oregon Mathematics Leadership Institute (OMLI) enabled a team of five instructors, including a master teacher and four mathematics faculty, to design and implement a course on non-Euclidean geometry for K-12 teachers. We focus on the intense structure of the course, which was delivered in 15 sessions of two hours each. We also discuss the content of the course, which included units on both taxicab and spherical geometry. Finally, we emphasize the pedagogy of the course, which included hands-on cooperative learning that was carefully orchestrated to ensure everyone’s participation, as well as skilled facilitation that was aimed at eliciting productive mathematics discourse, and thus at improving understanding of mathematical concepts.

THE NCTM NAVIGATING SERIES

**Maria G. Fung, Ph.D.**

We will present a selection of featured activities from the Navigating Series of the NCTM that focus on different grade levels and different topics in mathematics education.

WEIGHT AND GAS MILEAGE OF CARS

**Sarah Kendall and Joseph Geagea**

*Faculty Adviser: Maria Fung, Ph.D.*

We analyze data from two different variables, study their correlation, examine outliers, and analyze the validity of the linear model.

ARE NURSES’ SALARIES DEPENDENT ON THE PRICE OF LIVING?

**Lauren Lagace and Elizabeth Flight**

*Faculty Adviser: Maria Fung, Ph.D.*

We analyze data from two different variables, study their correlation, examine outliers, and analyze the validity of the linear model.

ACUTE TRIANGLE INEQUALITIES

**Kenneth Sanderson**

*Faculty Adviser: Hansun To, Ph.D.*

Mathematicians throughout the years have developed many powerful inequalities relating to acute triangles. One of the more famous is Euler’s Inequality, published in 1765, relating a triangle’s inradius to its circumradius. Consider an acute triangle with side-lengths a, b, and c, with inradius r and semiperimeter p. James Keenan, a 2008 WSC graduate, and I proved that (1-cosA)(1- cosB)(1- cosC) ≥ cosAcosBcosC(2- (3√3r)/p). In this PowerPoint presentation, I describe our proof, which was accomplished using Euler’s Inequality, Heron’s

Formula, Blundon’s Inequality and others.

CRYPTOGRAPHY, CREDIT CARDS, AND CELL PHONES

**Susan Schmoyer, Ph.D.**

Most of us use cryptography every day without even realizing it. Every time you use your cell phone and every time you buy a book online, you are using cryptography to send encrypted messages. In this poster we outline the basics of the cryptography involved and indicate how advanced mathematics is used in these everyday transactions.

THE TRIVIALITY AND NONTRIVIALITY OF TATE-LICHTENBAUM SELF PAIRINGS ON JACOBIANS OF CURVES

**Susan Schmoyer, Ph.D.**

Let E be an elliptic curve defined over a finite field F and suppose that E[n] is defined over F. For attacking the elliptic curve discrete logarithm problem it is useful to know when points pair with themselves nontrivially under the Tate-Lichtenbaum pairing. In this paper we characterize when all points in E[n] have trivial self pairings. This result is expressed in terms of the action of the Frobenius endomorphism on E[n^2]. We give examples of how this result can be used to derive some well-known residuacity laws. We then generalize the elliptic curve result to Jacobians of algebraic curves of arbitrary genus.

HOMOGENIZATION OF DYNAMIC LAMINATES

**Hansun To, Ph.D.**

This paper addresses the study of the homogenization problem associated with propagation of long wave disturbances in materials whose properties exhibit not only spatial but also temporal inhomogeneities (called dynamic materials). Homogenization theory is employed to replace an equation with oscillating coefficients by a homogenized equation. Two typical examples of periodic homogenization are considered: the wave equation and Maxwell’s system coefficients oscillating rapidly not only in space but also in time. Conditions that generate applicability of the homogenization procedure to dynamic materials composites are developed. The effective tensors of rank-one laminates for one-dimensional wave equation and the full Maxwell’s system are computed explicitly. We also note some dramatic differences between the hyperbolic and the elliptic cases.