CTY’s mathematics, science, and computer science courses are dedicated to Dr. Richard P. Longaker, Provost of Johns Hopkins University from 1979 to 1987, in recognition of his advocacy and guidance through CTY’s initial years. CTY’s mathematics courses offer students the opportunity to enrich or accelerate their study of mathematics. Students who wish to accelerate need to work closely in advance with their local schools to negotiate appropriate placement and/or credit. Please refer to our Eligibility web page 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 decided. 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 make the world in which we live a better place. Note: A graphing calculator, such as a TI-83, is recommended. Sample text: Excursions in Modern Mathematics, Arnold and Tannenbaum. Session 1: Saratoga Springs
Session 2: Saratoga Springs Top
Cryptology
Information is power. Even before the first written word, the need to safeguard information created an ongoing evolutionary battle between codemakers and codebreakers. Cryptology is the study of secret writing such as codes and ciphers. In this math course, students begin their journey with an exploration of many early techniques for creating secret writing, such as cipher wheels, the Caesar shift, monoalphabetic substitution, and the Vigenère cipher. They move on to learn about modern techniques including RSA public key cryptography. Delving deeper into modern techniques, students explore how data transmitted by computer can be secured with digital encryption. Discussions about the vulnerabilities of each encryption system enable students to attack and decrypt messages using techniques such as frequency analysis and cribbing. Students apply what they learn to encrypt and decrypt their own secret writing. Though the course’s central focus is on the mathematics of cryptology, the historical context of cryptography and cryptographic devices is provided to further develop understanding of this branch of mathematics. For example, students examine the design and fallibility of the Enigma Machine, one of the most important cryptographic devices in history. Sample text: The Code Book, Singh. Field Trip Fee (Carlisle and Lancaster only): $65. Students visit the National Security Agency’s National Cryptologic Museum. Session 1: Carlisle, Lancaster, Los Angeles, Loudonville
Session 2: Carlisle, Lancaster, Los Angeles, Loudonville Top
Probability and Game Theory
The study of probability and game theory is an excellent way for students to apply math to real-world situations. Unlike mathematical modeling, which is a broad field, game theory is a very specific branch of mathematics focusing on the application of probability to competitive behavior. Students investigate a variety of topics including voting patterns, coalition building, and bargaining. In this class, students use mathematical modeling as an important foundation from which they can pursue further investigations. The models of game theory are abstract representations of real-life situations. For example, the theory of Nash Equilibrium has been used to study political competition, and the Prisoner’s Dilemma has been used to analyze the social networks of different populations. The mathematics covered in this course includes concepts of probability and linear algebra. Class exercises involve individual and group work as well as possible class tournaments. Sample texts: Game Theory and Strategy, Straffin; Thinking Strategically, Dixit. Prerequisite: Algebra I. Session 1: Carlisle, Lancaster, Los Angeles, Saratoga Springs
Session 2: Carlisle, Lancaster, Los Angeles, Saratoga Springs Top
Mathematical Logic
Reasoning and logic form the backbone of almost every academic discipline. From philosophical argumentation to the scientific method, clear reasoning and consistent logic are the building blocks of serious inquiry. This course introduces students to the basics of mathematical logic and formal proof. Students acquire a foundation in formal logic, the basis of rigorous mathematical proof. They gain an understanding of the major elements of mathematical logic: validity, soundness, formal proof, and counterexample. As they explore the tools of reasoning, they develop strong problem-solving skills and learn to think analytically. This course emphasizes how to organize knowledge and present solutions to problems in a simple, coherent, and systematic manner. Students analyze the structure of proofs and solve a range of meta-mathematical problems. By the end of the course, students are able to translate statements and arguments and to write their own proofs with increased elegance and assurance. Sample texts: Logic: Techniques of Formal Reasoning, Kalish et al.; Formal Logic: Its Scope and Limits, Jeffrey. Prerequisite: Algebra I. Session 1: Baltimore, Lancaster
Session 2: Baltimore, Lancaster, Los Angeles Top
Number Theory
Called “the queen of mathematics” by Karl Friedrich Gauss, number theory is the study of the counting numbers, the most basic of all number systems. Despite the simplicity of the counting numbers, many accessible problems in number theory remain unsolved. For example, the Goldbach Conjecture formulated in 1742 posits that every even integer larger than 2 is the sum of two prime numbers. In this course, students are introduced to the major ideas of elementary number theory and the historical framework in which these concepts were developed. While strengthening their ability to analyze and construct formal proofs, students explore topics including the Euclidean Algorithm and continued fractions, Diophantine equations, Fibonacci numbers and the golden ratio, modular arithmetic, Fermat’s Little Theorem, RSA public key cryptography, and Fermat’s Two Square Theorem. Students leave the course with an appreciation of the elegance of theoretical mathematics. Sample text: Materials compiled by the instructor. Prerequisites: Geometry and Algebra II. Session 1: Lancaster
Session 2: Lancaster Top
Individually Paced Mathematics Sequence
Individually Paced Math Sequence includes courses in Algebra I, Geometry, Algebra II, and pre-calculus topics including Functions, Trigonometry, Discrete Math, and Analytic Geometry. It is designed for students with the interest and ability to accelerate their study of mathematics, perhaps to take Calculus at an earlier grade level or to begin studying college mathematics while still in high school. Inspired by research conducted by Johns Hopkins University’s Study of Mathematically Precocious Youth (SMPY), this course is designed for students with strong independent learning skills, high self-motivation, and high mathematical ability. Math Sequence allows students to work independently at a pace commensurate with their individual abilities. At the beginning of the session, instructors assess the knowledge base of each student with diagnostic testing. The results of this testing, together with information provided by students and their schools, determine the appropriate starting point for each student in CTY’s curriculum and allow students to focus their attention on unfamiliar material. For most of the day, students work on their own, solving problems from their textbooks. The instructor and teaching assistant monitor each student’s progress carefully and offer support and individualized instruction as needed. Students must demonstrate a mastery of the concepts and skills within each subject area before moving on to the next topic. For more information, visit the Math Sequence Web Page
Sample texts: Algebra, Smith; Geometry, Jurgensen; Algebra and Trigonometry, Smith; Advanced Mathematics, Brown; Precalculus with Trigonometry, Foerster. Session 1: Baltimore, Lancaster, Saratoga Springs
Session 2: Baltimore, Lancaster, Saratoga Springs Students who have not consulted with their home school regarding credit and placement should not register for this course. Top
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