# The subject that we casually refer to as algebra in fact encompasses an incredibly broad set of topics.

In the conventional math curriculum students are considered to have reached algebra once the concept of "variable" is introduced. However, the various sets of numbers with which our students have long been familiar (such as integers, fractions, or real numbers) are prime examples of algebraic structures. Furthermore, the properties of these sets of numbers are themselves worthy of study and can be extended to many more settings beyond just these particular number systems. Finally, it goes without saying that the traditional subject of algebra is an immensely powerful and versatile tool in the service of physics, engineering, economics, and much more.

Our suite of courses in Block 2 reflects all of these manifestations of algebra. During Algebra 1a and Algebra 2 students learn the tools of the trade and begin to see some of their applications. In the Axiomatic Development of Numbers course, students lay a rigorous foundation for the various number systems and develop their properties from scratch. Our Algebra 2 Lab course focuses on more sophisticated mathematical applications of algebra, while Intro to Ring Theory gives students a first look at the more abstract aspects of addition and multiplication, as they operate in contexts other than number systems. There's a lot of exciting math happening in our classrooms, as always!

*-- Sam Vandervelde*

**Algebra 1a**

with John DeIonno

This course provides a solid grounding in one of the most essential and ubiquitous tools used in mathematics: algebra. Students begin by understanding the concept of variable, then learn about manipulating algebraic expressions, solving equations, graphing lines, factoring to solve quadratic equations, and analyzing inequalities. The purpose of the course is to organically motivate the need for and utility of algebra, while providing adequate training in order to use algebra proficiently.

**Algebra 2 **

with Sam Vandervelde

This course continues the study of algebra, a subject on which nearly every other branch of mathematics depends. After a brief review of introductory concepts, students will study factoring, quadratic equations, graphs of circles and parabolas, linear changes of coordinates, polynomials and sequences. Together with the Algebra 1a/1b material, this class provides a complete introduction to the subject. In this course emphasis will be placed on individual proficiency in mastering skills, then applying them in a variety of settings.

**Algebra 2 Lab **

with Austin Shapiro

In this course, students who have already completed the introductory algebra sequence will conduct a series of week-long topical investigations, which serve to apply, reinforce, and extend their algebra skills. Topics in the first half of the course will center on algebraic inequalities and constrained optimization, such as isoperimetric problems. Topics in the second half will extend the concept of graphing to encompass conics, figures of revolution, and parametric curves. Emphasis will be on individual mastery of skills and on developing fluency in translating problems from geometry and other “languages” into algebra.

**Axiomatic Development of Numbers **

with Susan Durst

This course explores the axiomatic foundations of several algebraic structures that appear in many branches of mathematics. We will work with natural, rational, and complex numbers; along with polynomials, rational functions, and other number-like objects that have well-defined addition and multiplication operations. Students will become familiar with equivalence relations, learn what it means for an operation on equivalence classes to be well-defined, and see examples of isomorphisms and quotient structures. This course will emphasize writing short proofs based on formal definitions and axioms, but also include more substantial writing assignments based on group exploration.

**Introduction to Ring Theory**

with Sachi Hashimoto

This is a first course in abstract algebra, with a focus on rings. Emphasis will be placed on writing rigorous proofs and in developing comfort with increasingly abstract concepts. We will motivate the study of rings by looking at examples in number theory and geometry, which will give us intuition for rings as structures which model systems of numbers or functions. Course topics will include examples of rings, ring homomorphisms, ideals, quotient rings, and the Chinese Remainder Theorem. Time permitting, we will also study unique factorization domains, principle ideal domains, and the division algorithm.