Mathematics can be described as a language, a tool, a science, and an art. Computer Science is an area of study that continues to gain importance for its rich theory and wide applications to physical and social sciences. In CTY’s math and computer science courses, students move beyond basic skills to gain greater understanding of both the underlying principles and the intriguing ways these concepts can be applied and extended to a range of contexts.

Students have the opportunity to explore challenging material and strengthen their problem-solving skills. They investigate advanced concepts through a process of discovery and engagement that promotes a lifelong interest in the discipline. Through hands-on, thought-provoking exercises, students learn to make connections between abstract ideas and their uses in a range of fields, including science, engineering, economics, and advanced mathematics.

Please refer to our Eligibility page for minimum test score requirements for math courses. The following math courses are listed below:

- Paradoxes and Infinities
- Geometry through Art
- Mathematical Modeling
- The Mathematics of Money
- Game Theory and Economics
- Discrete Math
- Foundations of Programming

Sample syllabi for all courses are also available with each course description.

The second sentence is true. The first sentence is false. Are these sentences true or false? How is it that observing an orange pumpkin is seemingly evidence for the claim that all ravens are black? Students in this course explore conundrums like these as they analyze a range of mathematical and philosophical paradoxes.

Students begin 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 an appreciation for and insight into some of the most perplexing questions ever posed.

**Sample text:** Materials complied by the instructor.

**Session 1:** Bristol, Easton, Seattle

**Session 2:** Bristol, Easton, Santa Cruz, Seattle

“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. In this introductory geometry course, students learn about geometric figures, properties, and constructions, and 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 twentieth-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 of the mathematical aspects of art.

**Note:** Students who have taken CTY's Geometry and Its Applications class, or a high school geometry class, **should not** take this course.

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

**Sample text:** *Squaring the Circle: Geometry in Art and Architecture*, Calter.

**Session 1:** Haverford, Santa Cruz

**Session 2:** Haverford

Mathematics is more than just numbers and symbols on a page. Applications of mathematics are indispensable in the modern world. Math 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 how to create mathematical models to represent and solve problems across a broad range of disciplines, including political science, economics, biology, and physics.

Students in this class 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. While investigating 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*, Tannenbaum.

**Session 1:** Easton, Haverford, Santa Cruz

**Session 2:** Easton, Haverford, Santa Cruz

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 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, 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 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 also 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 sector.

**Sample text:** Materials compiled by the instructor.

**Session 1:** Bristol, Easton, Haverford, Santa Cruz

**Session 2:** Easton, Haverford, Santa Cruz

**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 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 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 equilibria 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 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 contexts in business and economics.

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

**Session 1:** Bristol, Santa Cruz

**Session 2:** Santa Cruz

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 answer this question in the affirmative, establishing the result known as the Four-Color Theorem. Discrete math introduces students to questions such as this as they learn math from a range of disciplines including set theory, combinatorics and graph theory, and number theory. This leads them to important real-world applications such as determining the number of ways to create a password of a given length or finding the shortest path for a taxi between multiple locations.

Students in this course begin by building a foundation in set theory and proof. They then explore combinatorics, examining the number of possible configurations of different sets of objects. Students move on to investigate graph theory, an area that introduces them to both historic problems such as the Bridges of Königsberg and the Traveling Salesman, as well as more modern applications such as the analysis of social networks and traffic patterns.

Students leave the course not only with a familiarity with a flourishing branch of mathematics, but also with an enriched mathematical vocabulary and an improved ability to understand and create mathematical arguments.

**Prerequisite:** Algebra I or a CTY Academic Explorations math course.

**Sample text:** Materials compiled by the instructor.

**Session 1:** Haverford

**Session 2:** Not offered

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 on 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 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 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 programming language learned may change based on the instructor’s preference.

**Sample text:** An introductory computer programming text.

**Lab Fee:** $65

**Session 1:** Bristol, Easton, Haverford

**Session 2:** Bristol, Easton, Haverford

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