List of Topics Addressed During the Semester

Math 496 Spring 2009

List of topics addressed during the semester

 Sequences and Series (including some time spent focusing on limits and whether or not .999 . . . = 1 {and how to help high school students understand why})

 Understanding Real numbers by linking them to Cauchy sequences

 Understanding the difference between a number and a number system. Concept analyses of number systems (natural numbers, integers, rational numbers, real numbers, complex numbers) and Abstract Algebra (Fields, Groups, Fundamental Theorem of Algebra, Rings, polynomials in F[x])

 Functions (Definition, inverse functions, compositions, restriction of functions, domain, co- domain, range, properties of functions). Later in the semester we also compared and contrasted Real functions and Complex functions. When we discussed Complex functions we connected back to the Taylor series

 Explain, contrast, compare, and give examples in the context of high school mathematics of the terminology such as equality, equivalence, non-equivalence, equivalent equations, equivalent relations and identities. We also tied this discussion back to isomorphic groups. These discussions also lead to a lesson about open & closed statements, equivalent equations, solution sets, compound sentences, systems of equations/inequalities, simultaneous equations, and zero product property.

 Geometry-related [using a compass and straightedge for basic constructions, discussing if it is possible to construct a line segment of any real-valued length, discussing un-constructible figures (i.e., doubling the cube, squaring the circle, and trisecting the angle)]

 Deriving and applying a variety of basic facts and relationships from Trigonometry (definition of and measurement of angles, properties of circles, deriving Trig identities using a circle of radius 1 or a unit circle, using a dynamic technological tool to represent the derivation of some compound identities such as Sin(2A)

 Applying the trig identities to prove why particular constructions with straight edge and compass do or do not work

 Making connections between Geometry and Algebra by looking at geometric constructions as field operations with the set of constructible lengths . The Geometric constructions were viewed as motions in space by discussing linear mappings and vectors in . And connections were made back to matrices, polar coordinates, complex numbers.

 Toward the end of the semester we tied back to earlier discussions of unions & intersections of sets, and focused on countably infinite sets—we discussed whether particular sets (natural numbers, integers, algebraic numbers, set of polynomials of degree n in Q[x], real numbers or complex numbers) were countable