Mathematics can be described as a language, tool, science, and art. CTY math courses aim to give students the skills, knowledge, and perspective to understand the multifaceted nature of mathematics and by doing so, enrich and accelerate their explorations of advanced math content. Students move beyond basic skills to gain greater understanding of both the underlying meanings of mathematical concepts and the intriguing ways math can be applied and extended in a range of contexts. CAA mathematics courses and computer science courses offer students the opportunity to strengthen their problem-solving skills and explore challenging material. Students investigate advanced mathematic concepts through a process of discovery and engagement that promotes a lifelong interest in the discipline. These courses ask students not only to master concepts and theoretical material, but also to apply their new knowledge to real-world questions and problems. Through hands-on approaches and thought-provoking exercises, students learn to make connections between abstract ideas and their use in a range of fields, including science, engineering, economics, and advanced mathematics. Please refer to the Eligibility section of this catalog for minimum test score requirements for math courses. Sample syllabi for all courses are also available.
Mathematical Modeling
Mathematics is more than just numbers and symbols on a page. It can be used to determine whether a meteor will impact Earth, predict the spread of an infectious disease, or analyze a remarkably close presidential election. Applications of mathematics are indispensable in the modern world. In this course, students learn how to create mathematical models to represent and solve problems across a broad range of disciplines, including political science, economics, biology, and physics. Students investigate voting systems by constructing mathematical models of how groups make decisions and how elections are conducted. They consider how goods, property, and even political power can be fairly divided and apportioned. Students learn how to use Euler and Hamilton circuits to find the optimal solutions in a variety of real-world situations, such as determining the most efficient way to schedule airline travel. In investigations of growth and symmetry, students develop linear and exponential growth models and explore fractals and the Fibonacci numbers. Students leave this course with the ability to use the seemingly abstract language of mathematics to gain a greater understanding of the world around them. Note: A graphing calculator, such as a TI-83, is recommended. Sample text: Excursions in Modern Mathematics, Arnold and Tannenbaum. Session 1: Easton, Santa Cruz
Session 2: Easton, Santa Cruz Top
The Mathematics of Money
From managing one’s personal investments to examining the profitability of a multibillion-dollar global corporation, the mathematics of money is at the heart of successful financial endeavors. Why are round-trip fares from Orlando to Kansas City higher than those from Kansas City to Orlando? How do interest rate adjustments made by the Federal Reserve affect the real estate market? How does one calculate the price-earnings ratio of a stock and use that to help predict that stock’s future performance? Mathematics is an indispensable part of the answer to each of these questions. This course provides students with a rigorous mathematical grounding in central concepts of business and finance. Students investigate the mathematics of buying and selling, and apply these principles to real world situations. They gain fluency with the concepts of simple and compound interest and learn how these affect the present and future value of loans, mortgages, and interest-bearing accounts. Students also explore stocks and bonds and acquire a firm understanding of national and international financial markets. In their examination of these topics, students manipulate and solve algebraic expressions, and also learn to apply a range of mathematical concepts including direct and indirect variation and arithmetic and exponential growth. Through simulations, projects, and classroom investigations, this course not only provides students with the foundation required to be more secure in their own personal financial management, and enhance their understanding of the broader economic conditions that shape those investments. Sample texts: Business Mathematics, Miller et al; A Beginner’s Guide to the World Economy, Epping. Session 1: Bethlehem, Bristol, Santa Cruz
Session 2: Bethlehem, Bristol, Santa Cruz
Top
Game Theory and Economics Prerequisite: Algebra I. Thomas J. Watson, the founder of IBM, once said, “Business is a game—the greatest game in the world if you know how to play it.” In today’s global marketplace, understanding game theory, the branch of mathematics which focuses on the application of probability to competitive behavior, is crucial to understanding business and economics. In this course, students use game theory as a framework from which to analyze a variety of real-world economic situations. Students begin the course by analyzing simple games, such as two-person, zero-sum games, and learn how these games can be used to model actual situations encountered by entrepreneurs and economists. For instance, Students may apply the concept of Nash equilibrium to find the optimum strategy for the pricing of pizza in the competition between Domino’s and Pizza Hut. As they acquire an understanding of more complex games, students apply these theories to analyze a variety of economic situations, which may include auctions and bidding behavior, fair division and profit sharing, monopolies and oligopolies, and bankruptcy. Through class discussions, activities, research, and mathematical analysis, students learn to predict and understand human behavior in a variety of real-world contexts in business and economics. Sample text: New course. Session 1: Bristol, Santa Cruz
Session 2: Bethlehem, Santa Cruz
Geometry and Its Applications
The word “geometry” means “earth measurement.” It is the branch of mathematics most strongly connected to the physical world. Therefore, it has many applications to problems encountered in everyday life by every culture, past and present. Long before Pythagoras presented his famous theorem, ancient Egyptians used geometry to form right angles to resurvey the Nile River Valley after the annual floods. Today, NASA scientists use the same formulas and theorems to determine the proper angles and arcs for the orbital paths of modern telecommunications satellites. In this course, designed as an introduction to geometry, students learn about geometric figures, properties, constructions, and proofs, with an emphasis on their wide applicability to human activity. Concepts are studied in depth through practice exercises and problem-solving activities. Students are exposed to many examples in which geometry is used in recreation, practical tasks, science, and the arts. While this course covers conceptual material, the focus is on applying geometry to solve problems and on discovering the importance of mathematics in a wide range of disciplines and situations. Note: This course is recommended as preparation for students who plan to take Geometry at school. Students should not use this course to replace a geometry course at their schools. Students are exposed to geometric properties and concepts, but due to the focus on applications they do not cover the breadth of material found in a year-long geometry course. Sample text: Discovering Geometry: An Inductive Approach, Serra. Session 1: Easton, Santa Cruz
Session 2: Easton Students who have already taken Geometry should not take this course. Top
Discrete Math
Prerequisite: Algebra I or a CAA math course. Can any given map be colored with just four different colors such that no two regions sharing a common edge are the same color? Mathematicians took more than 100 years to demonstrate this is possible and thus prove the Four-Color Theorem. Such questions are the domain of discrete math, a field that examines objects that have values characterized by natural numbers. This course focuses on two core areas within discrete math: combinatorics and graph theory. Students begin by building a foundation in set theory, induction, and proof construction. They then explore combinatorics, examining the number of possible configurations of different sets of objects. Even when the conditions specified for a configuration are relatively simple, counting the possibilities often requires great ingenuity. Combinatorics lays the groundwork for investigations in diverse areas of higher mathematics, including probability, optimization, and number theory. Unlike familiar graphs of equations, graph theory focuses on objects and the connections between them. Students discover natural applications for enumeration techniques when they delve into graph theory, a subject rich in abstract concepts, such as counting and coloring theorems, and in applications, such as traffic networks, models for molecules, and link structures for websites. Students leave the course with an enriched mathematical vocabulary, a familiarity with a flourishing branch of mathematics, and an ability to understand and create original mathematical arguments. Sample text: Materials compiled by the instructor. Session 1: Easton
Session 2: Not offered Top
Chaos and Fractals
Prerequisite: Algebra I. Can a small action in one part of the world lead to catastrophic consequences in another? Can order be revealed in chaos? For mathematicians, the quest to understand and structure the unpredictable dates back to the 1890s and the renowned French mathematician Henri Poincaré. Chaos theory today is an important area of study with a wide range of applications in the physical and social sciences as well as the arts. In this course, students investigate the mathematical foundations of chaotic dynamical systems and fractals. They begin by learning the fundamental process of iteration in functions. With that foundation, they examine more advanced concepts including bifurcations and sensitive dependence. Linking this abstract mathematics with the real world, students explore applications of chaos theory such as forecasting weather, understanding purchasing power and world markets, or examining heart arrhythmia. Students apply their knowledge of iterated functions and chaotic systems to investigate the origins and applications of fractals, geometric patterns which exhibit some degree of self-similarity at any scale. Such fractals may include the Cantor set and the Mandelbrot set. Through these topics students gain a more formal understanding of the basic principles of chaos theory, an area of math usually studied only by undergraduate and graduate students. Sample text: Materials compiled by the instructor. Session 1: Not offered
Session 2: Easton Top
Foundations of Programming
Students in this course gain insight into the methods of computer programming and have an opportunity to study the algorithmic aspects of computer science. They learn the theoretical constructs common to all languages by studying the syntax and basic commands of a particular programming language. Building on this knowledge, students move on to study additional concepts of programming, including arrays, testing, and debugging. Students are given a variety of challenging problems which teach them to start with a concept and work through the steps of writing a program: defining the problem and its desired solution, outlining an approach, encoding the algorithm, and debugging the code. Through a combination of individual and group work, students complete supplemental problems, lab exercises, and various programming projects in order to reinforce concepts learned in class. By the end of the course, students are able to develop more complex programs and are familiar with some of the standards of software development practiced in the professional world. Students leave with an understanding of how to apply the techniques learned to other programming languages. Note: Students who have taken CAA’s Computer Science: Introduction to Programming should not take this course. Sample text: An introductory computer programming text. Lab Fee: $65 Session 1: Bethlehem, Bristol, Easton
Session 2: Bethlehem, Bristol, Easton Top
| 
