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.

Economics is an essential tool for understanding the underpinnings of modern society. In CTY’s economics courses, students gain an understanding of the central concepts behind trade and finance on both the local and international scale.

In all of these courses, students have the opportunity to enrich their studies. They investigate advanced concepts through a process of discovery and engagement that promotes a lifelong interest in these disciplines. Through hands-on, thought-provoking exercises, students learn to make connections between abstract ideas and their use in a range of fields, including science, engineering, and advanced mathematics.

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

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.

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

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

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

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

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

"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, 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 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 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:** Materials compiled by the instructor.

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

- Sample syllabus (PDF)

**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 TI-84, is required.

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

**Session 1:** Allentown**Session 2:** Haverford

- Sample syllabus (PDF)

**Prerequisite:** Algebra I

During the CTY all-site meeting on opening day, what is the smallest number of students who need to enter the auditorium before there will be at least three students in the room who already knew each other before attending CTY, or at least three students who were all strangers before they arrived? This problem can be illustrated by a type of graph in which vertices representing students are connected by edges that are colored to indicate friendships. This course introduces students to such problems and how to approach them as they learn the mathematics of combinatorics.

This course begins with an exploration of enumerative combinatorics. Students use different techniques of mathematical proof and ideas in set theory to solve a variety of problems, and they study topics such as binomial coefficients, permutations, and partitions. For example, students employ bijective functions to turn seemingly tedious counting problems into much simpler problems, such as how to determine the number of games played in the March Madness tournament without needing to pull out your bracket and count one by one.

Students then investigate graph theory, an area of mathematics that is used in modern applications in fields such as computer science, counterterrorism, and navigation. One famous question in graph theory posed in the early 1800s—whether you can color any map using just four colors so that no two adjacent areas share the same color—took more than 100 years for mathematicians to answer in the affirmative. Students explore this question and other historic problems in graph theory as they delve into concepts such as cycles, planarity, algorithms, and graph colorings.

Along the way, students encounter problems that are challenging and fun, and that can be solved with creativity and determination, even without a prior background in college-level math.

**Note:** Students who have taken CTY's Discrete Math course should not take this course.

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

**Session 1:** Not offered**Session 2:** Santa Cruz

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

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 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

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

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

- Sample syllabus 1 (PDF)
- Sample syllabus 2 (PDF)

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

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