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Math and Computer Science Courses

In CTY’s math, computer science, and economics courses, students move beyond basic skills to gain greater understanding of both the underlying principles and the intriguing ways these disciplines can be applied to a range of contexts.

Please refer to our Eligibility page for minimum test score requirements for math courses.

Mathematics

Computer Science

Economics

Please note that the sample syllabi are meant to provide an idea of the level of the course and whether or not the content area interests you. CTY instructors are given guidelines within which they each develop their own syllabi. Choice and sequencing of topics within the content area, as well as specific activities, labs, and assignments, will vary.

Paradoxes and Infinities

Students in this course explore conundrums as they analyze a range of mathematical and philosophical paradoxes. They begin this course by considering Zeno’s paradoxes of space and time, such as The Racecourse in which Achilles continually travels half of the remaining distance and so seemingly can never reach the finish line. To address this class of paradoxes, students are introduced to the concepts of infinite series and limits. Students also explore paradoxes of set theory, self-reference, and truth, such as Russell’s Paradox, which asks: Who shaves a barber who shaves all and only those who do not shave themselves? Students analyze the Paradox of the Ravens as they study paradoxes of probability and inductive reasoning. Finally, they examine the concept of infinity and its paradoxes and demonstrate that some infinities are bigger than others.

Through their investigations, students acquire skills and concepts that are foundational for higher-level mathematics. Students learn and apply the basics of set theory, logic, and mathematical proof. They leave the course with more nuanced problem-solving skills, an enriched mathematical vocabulary, and insight into some of the most perplexing questions ever posed.

Sample text: Materials compiled by the instructor.

Session 1: Allentown, Bristol, Santa Cruz, Seattle
Session 2: Allentown, Bristol, Santa Cruz, Seattle

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Geometry through Art

"Geometry is the right foundation of all painting." In this way, the German artist Albrecht Dürer described a connection between mathematics and art that can be found in every culture. Students in this introductory geometry course learn about geometric figures, properties, and constructions, then use this knowledge to analyze works of art ranging from ancient Greek statues to the modern art of Salvador Dalí.

Beginning with the foundations of Euclidean geometry, including lines, angles, triangles, and other polygons, students examine tessellations and two-dimensional symmetry. Using what they learn about points, lines, and planes, students investigate the development of perspective in Renaissance art. Next, they venture into three dimensions, analyzing the geometry of polyhedra and considering their place in ancient art. Finally, students explore non-Euclidean geometry and its links to 20th-century art, including the drawings of M. C. Escher.

Through lectures, discussions, hands-on modeling, and small group work, students gain a strong foundation for the further study of geometry, as well as an appreciation for the mathematical aspects of art.

Note: Students who have taken a high school geometry class should not take this course.

Note: This course exposes students to geometric properties and concepts, but it should not be used to substitute a year-long high school geometry course.

Sample text: Materials compiled by the instructor.

Session 1: Haverford, Santa Cruz
Session 2: Haverford, Santa Cruz

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Mathematical Modeling

Prerequisite: Algebra I

Mathematics transcends numbers and symbols on a page. We observe it in nature, social behavior, and even in the way a population grows. Viewing mathematics through these lenses is indispensable in the modern world. Math, for example, 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. In this course, students learn, create, analyze, review, and evaluate mathematical models to represent and solve problems across a broad range of disciplines, including political science, economics, biology, medicine, and physics.

Students begin with a review of some of the core mathematical tools in modeling, such as linear functions, lines of best fit, and exponential and logarithmic functions. Using these tools, students examine models such as those used in population growth and decay, voting systems, or the motion of a spring. Students also 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. A review of probability may lead into a study of using deterministic versus stochastic models to predict the spread of an epidemic. Students leave this course familiar with all steps of the modeling process, from defining the problem and making assumptions to assessing the model for strengths and weaknesses.

Note: A graphing calculator, such as a TI-83 Plus or a TI-84, is required.

Sample text: Materials compiled by the instructor.

Session 1: Allentown
Session 2: Haverford

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Foundations of Programming

Students in this course gain insight into methods of computer programming and explore the algorithmic aspects of computer science. They learn the theoretical constructs common to all high-level programming languages by studying the syntax and basic commands of a particular programming language such as Java, C, C++, or Python. Building upon this knowledge, students move on to study additional concepts of programming, such as object-oriented programming or graphical user interfaces. By solving a variety of challenging problems, students learn to start with a concept and work through the steps of writing a program: defining a 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 to reinforce concepts learned in class. By the end of the course, students can 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 high-level programming languages.

Note: The specific programming language used is based on the instructor’s preference.

Sample text: An introductory computer programming text.

Lab Fee: $135

Session 1: Allentown, Bristol, Haverford
Session 2: Allentown, Bristol, Haverford

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The Mathematics of Money

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-to-earnings ratio of a stock and use that result to help predict that stock’s future performance? Mathematics plays an indispensable part in answering each of these questions.

This course provides students with a mathematical grounding in central concepts of business and finance. Students investigate the mathematics of buying and selling, then 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 investigate various forms of taxes, considering their impact on personal and governmental budgets. In their examination of these topics, students manipulate and solve algebraic expressions and learn to apply a range of mathematical concepts, including direct and indirect variation and arithmetic and exponential growth. Through simulations, entrepreneurial projects, and classroom investigations, this course provides students with the foundation required to be more secure in their financial management and enhances their understanding of the broader economic conditions that shape investments in the public and private sectors.

Sample text: Materials compiled by the instructor.

Session 1: Allentown, Bristol, Haverford, Santa Cruz, Seattle
Session 2: Allentown, Bristol, Haverford, Hong Kong, Santa Cruz, Seattle

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Game Theory and Economics

Prerequisite: Algebra I

Thomas J. Watson, 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 that focuses on the application of strategic reasoning 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 learning how they can be used to model actual situations encountered by entrepreneurs and economists. For instance, students may apply the concept of Nash equilibria to find the optimal strategy for pricing pizza in the competition between Domino’s and Pizza Hut.

As they acquire an understanding of more complex games, students apply these methods 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 business and economics contexts.

Sample texts: Game Theory and Strategy, Straffin; Thinking Strategically, Dixit and Nalebuff.

Session 1: Bristol, Santa Cruz
Session 2: Bristol, Hong Kong, Santa Cruz

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