Algebra 1 (8th Grade)
In Algebra 1 students learn to generalize the laws of arithmetic and perform the four operations on variable expressions. They develop their ability to model and solve word problems by assigning variables to unknown quantities and determining the precise relationship between constant and variable terms. Students apply the laws of equality in order to solve a wide variety of equations and proportions. In the process of graphing the solution sets of linear equations on the Cartesian plane, students gain familiarity with the concepts of slope and intercept. They find simultaneous solutions to systems of equations and apply factoring in order to find the roots of quadratic equations. All of these activities promote both arithmetic and algebraic fluency.
In Geometry, we study the world of points, lines, and planes. We cover topics that include the analysis of congruent and similar triangles, the Pythagorean Theorem, angle sum and area formulas, and theorems concerning the relationship between chords, secants, and tangents of a circle. We solve problems and explore geometric situations intuitively; we also investigate geometry as a formal system, where we begin with a small set of postulates and then build up a Euclidean geometric system by deductively proving further results. With this balance, we uncover mathematics the way it often plays out historically, where bursts of intuition drive knowledge forward, and then formalization solidifies known results into a cohesive whole.
Students come to Algebra 2 after having had a year of Geometry, and this knowledge is applied on a regular basis. The Cartesian plane provides a setting for examining transformations such as reflection, translation and scaling. Parallel and perpendicular lines are analyzed using the concept of slope. Functions are examined both algebraically and graphically, as are systems of equations and inequalities. Students also work in a purely algebraic setting, solving equations, manipulating algebraic expressions, working with higher-degree polynomials, expanding binomial powers, and examining rational expressions. The challenge of solving quadratic equations leads to such techniques as factoring, completing the square, the quadratic formula, and the discovery of the complex numbers.
Electives: Full-Year Courses
This is a rigorous approach to polynomial, trigonometric, and exponential functions: sequences and series; vectors; and some analytic geometry. Emphasis is on the mastery of proofs and creative applications to practical problems. This course is a prerequisite for calculus.
Prerequisite: Algebra 2
This is a rigorous calculus course with heavy emphasis on proofs, derivations, and creative applications. Limits, derivatives, integrals, and their technical applications are covered. This course will include an early use of transcendental functions and will require a working knowledge of trigonometric, exponential, logarithmic, and rational functions.
Calculus 2 is a continuation and expansion of the techniques of Calculus 1. It includes a review and a proof of the fundamental theorem of Calculus, further methods of integration with application to physical problems, alternative coordinate systems, series and sequences, vector functions, and differential equations.
Prerequisite: Calculus 1
Advanced Problem Solving (2x per week)
This course is designed for students intending to participate in the school’s math team. We focus on mathematical topics not typically covered in other course. Topics such as number theory and modular arithmetic, combinatorics, polynomials, geometric loci, probability, functional equations, algebraic and trigonometric identities, geometric inequalities, divisibility, colorings and tilings, Diophantine equations, three dimensional geometry, complex numbers, recursions, infinite series, graph theory, quadratic forms, abstract algebra, generating functions, geometry of conic sections, optimization, spherical trigonometry and logic are explored through a series of problems, often selected from various mathematical contests. We meet twice a week to discuss the solutions to problem sets we’ve been working on over the previous week. Students are encouraged to conduct outside research and present their discoveries at an in-school math fair.
Prerequisite: Algebra 2, or permission of instructor.
Game Theory 101
How do hawks coordinate their hunt? How does a stallion decide when to fight and when to back down? How do apes decide when to share, whom to trust, whom to deceive? How do entire lineages decide how much energy to expend on nurturing the young? When we sit down at the poker table, how do we formulate a betting strategy? Does it change fluidly in response to the behavior of others at the table? Is there any way to model such a thing, or are we stuck with our “gut” intuition? When we allow contractors to bid for that prestigious linoleum-countertop contract, when we decline the steroids even as we suspect others are benefiting from them, when we consider evolving a new limb over the next million years, when we form alliances with countries (or species) we can’t entirely trust… WHAT ARE WE GETTING OURSELVES INTO?!?
There’s no better way to develop a deep understanding of these multifarious scenarios than to actually PLAY the GAMES! We will spend our time developing game-theoretic models for everything from card games to ecosystems, from financial markets to dating strategies, and testing them in the lab of our own classroom. While we will be dealing on a deep level with very complex systems, there won’t be too much formalism (“math”) — we evaluate our games according to how well they model real-world scenarios, and how simple, fun, and enlightening they are to play.
Math Of Life
P. Theodosopoulos, T. Theodosopoulos
The intersection of biology and mathematics is an amazing frontier! Enter the realm of how living systems evolved and work together to create a functioning whole. Join us in Math of Life as we explore the emerging field of mathematical biology. We ponder the big questions and build models to understand the complexity of biological systems. The modeling process plays a central role in this class, offering opportunities to study various mathematical concepts in context, including dynamical systems, Markov chains, random walks and optimization. Students help design and run laboratory and computer-based experiments to illustrate these processes in biological systems, and practice statistical analysis and interpretation of the results.
This year we will focus on two topics, genetics and epigenetics in the fall term, and epidemiology and immunology in the spring term. The exciting new field of epigenetic dynamics extends population genetics, getting to the heart of the “nature vs. nurture” debate. Epigenetic modification to our DNA appears to underlie the expression of traits in both healthy and disease states as diverse as cancer, obesity and autism. Epidemiology attempts to track the development and spread of dangerous emerging pathogens through spatial diffusion and mutation models. These pathogens evolve along with our immune system. Did you know that whenever you are exposed to a new pathogen some of your immune cells are actually permitted, even encouraged, to mutate their DNA? Amazing, but true, and mathematical models are leading the way to understanding the how and why. That’s the math of life.
You can find more information about our Math of Life class, including our physical and computational labs, presentations and our reading and writing assignments, on our website, https://sites.google.com/a/saintannsny.org/mathematics-of-life/.
Prerequisite: Biology and Algebra 2
Patterns Through Math and Music
(Please see Interdisciplinary Studies)
Principles Of Economics 1 (Fall semester)
This is the first part of a course that introduces the economic sphere, its questions, and methods of inquiry. Topics to be covered include the theory of supply and demand, market equilibrium, aggregate measures of economic health such as the gross domestic product, the short-term equilibration of the goods and money markets, monetary and fiscal policy making, the IS/LM equilibrium and the theory of the consumer, including a study of utilitarianism and its criticisms. Students present weekly news reports on current topics of interest, probing the political and social implications of economic decisions. Selected readings from the economic literature and in-class policy design and writing exercises lead to two papers, probing the shortcomings of GDP and the limitations of the representative agent model.
Prerequisite: Algebra 2
Principles Of Economics 2 (Spring semester)
This is the second part of a course that introduces the economic sphere, its questions and methods of inquiry. Topics to be covered include the labor market and industrial relations, unemployment and inflation, the Phillips curve and the medium term general equilibrium, the theory of the firm, cost and production functions, scale and scope decisions, and market mechanism design. Students continue presenting weekly news reports on current topics of interest, probing the political and social implications of economic decisions. A study of macroeconomic policy-making leads to a weeklong computer simulation of the national economy, and a paper describing the challenges faced by a central banker. The class culminates with a month-long study of selections from the economic literature, ranging across extension topics from political economy, behavioral economics, and sociology. These readings prepare us for the annual Economics Debate, a moderated event, open to the larger Saint Ann’s community, focusing on a topic of current interest.
Prerequisite: Algebra 2. Note: Principles of Economics 1 is not a prerequisite for this course.
Is Derek Jeter clutch? Does Carmelo Anthony take too many shots? How can we write an algorithm for ranking tennis players? Should you go for it on 4th and 2? When is it time to call for a relief pitcher? How often do Olympic judges meaningfully disagree and how many judges do we really need? How should I fill out my NCAA bracket?
In this class we will question conventional sports wisdom at every turn. We will do so largely by examining and analyzing data – calculating correlations, visualizing data, performing regressions and competing to make the best predictions. We will discuss the major findings in sabermetrics and APBRmetrics and explore normal, binomial and Poisson distributions. Each student will also have the opportunity to research a sport and a question of particular interest to him/her.
Prerequisite: An interest in sports or data that extends past mere fandom and into the nerdy depths that you don’t talk about in polite company.
Independent Study in Mathematics
Students work one-on-one with a mentor, on a focused research project. Topics are to be determined by interest and inclination of the student. Students who enroll in Independent Study are also enrolled in a weekly research forum with other independent study students. this weekly meeting provides a venue for sharing research questions, discussing recent mathematical discoveries, and practicing writing and presenting mathematical ideas. Activities vary widely during the year, from reading and discussing mathematical literature, to presenting the students’ independent work, to working together as a group to investigate sample research questions, both historical and contemporary.
Prerequisite: Students must submit a one page proposal outlining their potential research interest in order to be considered for this course. This should be given to the High School office or the Math department by June 1st.
Electives: One Semester Courses
The Mathematics Department has developed the following set of one-semester electives to add to our choices of high school courses. Depending on enrollment, not all electives will necessarily run next year. If you register for one of these electives, please be prepared to select an alternate choice. Depending on interest, courses that do not run next year may run the following year and become part of an alternate-year cycle.
Formal Logic(Fall semester)
Formal logic, a discipline created by Aristotle, has applications in a variety of disciplines including philosophy, mathematics, physics, computer science and linguistics. One might in fact argue that logic is relevant to any endeavor that involves reasoning. This course begins with a consideration of arguments of English and the question: What constitutes a good argument? We then focus on the symbolic system known as sentential logic and the more powerful symbolic system knows as predicate logic. In both cases, students learn to translate arguments of English into symbolic arguments and to evaluate such arguments using the aforementioned systems. This is a proof intensive class.
Number Theory (Fall semester)
What did Leopold Kronecker mean when he famously said, “God made the natural numbers; all else is the work of man?” Even those who take the more humanistic stance that “math is created” rather than “math is discovered,” sometimes admit that the counting numbers seem to transcend humankind. How could we have invented “1, 2, 3, 4, 5…?” Even more compelling, and somewhat surprising, is that this seemingly simple set of numbers provides mathematicians with some of the deepest problems. Many of these questions are so simple that they can be understood by a five year old, yet many remain unsolved by anyone on Earth. In this course, we will study this set of numbers as number theorists do. We look for attractive properties and patterns, attempt to describe them, ask questions, venture answers, and, when we can, prove our conjectures. We will explore questions that range from the elementary, “Which combinations of integers make Pythagorean triples?” to the more difficult, “When is a number the sum of two squares?” to the unsolved, “Are all even numbers the sum of two primes?” and we will see that the most elementary set of numbers provides enough depth to leave us inspired, puzzled, bemused, and humbled.
Prerequisite: No official prerequisites – Number Theory is about elementary arithmetic, after all. But it gets deep and gnarly quickly! Only those who won’t balk at conceptually mind-bending arguments and technically demanding swamplands should take this course.
Symmetry (Spring semester)
What do we mean when we say that an object or design is symmetrical? We mean that there is some sort of action (such as reflection or rotation) that can be performed on the thing that leaves it unchanged. Group Theory begins as the study of such structure-preserving transformations and the ways they combine and interact to form closed systems or “groups” of symmetries. The study of symmetry in the abstract has proved to be a powerful tool in the investigation of a wide variety of problems throughout modern mathematics and theoretical physics. Symmetry groups are naturally classified by size and complexity, leading to many fascinating and difficult problems that lie at the heart of our ideas of mathematical beauty.
Prerequisite: Permission of the Department
Trigonometry (Fall semester)
Beginning with trigonometric functions and triangle solutions, we move on to identities, equations, angle formulae, and the practical applications thereof. Last, we cover the graphs of all the trigonometric functions including inverses and period, amplitude, and phase shifts.
Prerequisite: Algebra 2