# Mathematics

#### A Visual Approach to Mathematics

This presentation emphasizes how mathematics educators should be presenting material visually to students so that they can understand why analytical and algebraic procedures work, demonstrates what teachers should be conscious of while preparing their lessons, and will shed light on a perspective of mathematics teaching that is often ignored.

#### Arbelos and Their Generalizations

An arbelos is a flat region bounded by the union of three tangent semicircles whose diameters all lie on the same line. This presentation will explore the properties of the arbelos and look at the properties of other figures called f-belos bounded by similar curves such as parabolas.

#### Introduction to Knot Theory

A mathematical knot is a curve that begins and ends at the same point and does not intersect itself at any point. This talk is an introduction to the mathematical theory of knots with an emphasis identifying and distinguishing different knots and some recent applications in Biology and Astronomy.

#### Overlapping Squares Puzzle

A puzzle, entitled “Eight Squares,” requires the reader to determine the order of eight squares of the same dimensions in a designated grid. This talk uses the idea of this puzzle, and looks at how one can determine the order of any number of three-by-three squares in a five-by-five grid.

#### Soddy Triangles

Three circles are configured so that each pair has exactly one point in common and there is exactly one line that is tangent to all three. A Soddy triangle is the triangle with the centers of the circles as vertices. This presentation will investigate properties of these triangles.

#### Square Triangular Numbers

A perfect square number can be written in the form m^2 for some natural number m. A triangular number can be written in the form n(n+1)/2 for some natural number n. A square triangular number is both square and triangular. This presentation will show interesting properties of square triangular numbers.

#### Symmetries of a Square and Pentagon

This presentation will develop the symmetries of a square and a regular pentagon through an algebraic perspective and generalize this process to an n-sided polygon.

#### The Kolakoski Sequence

The Kolakoski sequence is an infinite, self determined sequence of 1’s and 2’s. We will define this sequence in such a way that it is clear what is meant by "self-determined" and we will discuss geometric ways of representing the sequence as well as applications in computer science.

#### The Tales of Textbooks

Educators of mathematics in the United States rely on textbooks for subject and pedagogical knowledge; this presentation will analyze the content and layout of textbooks from three pivotal eras to aid in defining the various stages of mathematics education throughout history.

#### Tossing a Coin Over the Telephone

There are many examples in social situations when flipping a coin is used as a fair way to determine who might buy dinner. However if two people on the phone need to make that decision, then flipping a coin won’t work because each person will think the other one will cheat. This talk will analyze the math behind flipping a coin on the phone so that cheating cannot occur.