This ALGEBRA I: TECHNIQUES AND APPLICATIONS (8th grade)

In this presentation of the Algebra I curriculum, 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.

ALGEBRA I (8th grade)

This course traces the historical, philosophical, and aesthetic development of the subject from ancient Babylonian problem tablets and Egyptian number puzzles to the high art of the Renaissance algebraists. Our survey of classical algebra will include the study of linear and quadratic systems, polynomials, roots and factorizations, the complex numbers, and elementary algebraic geometry. Students will engage firsthand as mathematicians—posing and solving their own problems, creating and developing their own techniques and problem solving strategies, and working together as a mathematical community. Our in-class math journal will provide students with an opportunity to share ideas, critique each other’s work, and to develop their own personal mathematical expository style.

GEOMETRY

All geometry courses will 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. All geometry courses will also, though to differing degrees construct proofs, solve geometric problems, and explore geometric patterns. With advisement, students will choose one of the three following:

• GEOMETRY: DEDUCTIVE SYSTEM BUILDING

In this course, we begin with a small set of postulates, properties that have overpowering intuitive appeal, and then we build up a Euclidean geometric system by deductively proving further results. Along the way, we investigate and employ a variety of proof techniques- regularly looking at alternate proofs of theorems.

• GEOMETRY: INDUCTIVE INVESTIGATIONS

In this course, we start our geometric inquiries in the middle of things. Specific problems serve as a starting point as we organize and extend our prior knowledge about shapes and their properties. By doing calculations and searching for patterns, students inductively formulate conjectures that are justified by previous results. Ultimately, we come to an understanding of Euclidean geometry as a cohesive whole.

• GEOMETRY: PROBLEM SOLVING

In this course, we begin with major results of geometry fully formed. After gaining an understanding of what these properties mean, we then use them to solve a spectrum of different types of problems- some straightforward, some demanding a great deal of insight. In so doing, we will naturally deepen our grasp of the properties with which we began and their interconnections.

ALGEBRA II

With advisement, students will choose one of the two following:

• ALGEBRA II: ANALYTIC GEOMETRY

How can geometric forms be characterized algebraically? This presentation of the Algebra II curriculum aims to synthesize the algebraic and geometric viewpoints. Parallel and perpendicular lines are analyzed using the concept of slope. Geometric transformations such as reflection and translation and scaling are explored by graphing sets of equations on the Cartesian plane. Theorems in geometry concerning similar figures, right triangles, and properties of a circle are handled using algebraic equations. The quadratic formula is derived and the roots of the second-degree equation lead to the discovery of complex numbers and the complex plane. The binomial expansion allows students to characterize the relationship between counting techniques and the coefficients in Pascal’s triangle.

• ALGEBRA II: FUNCTIONS AND ABSTRACT ALGEBRA

What is an algebra in its essential form, and when are equations solvable? This presentation of the Algebra II curriculum will answer these questions and explore their implications. Input/output machines called functions will be around every corner- as an object for investigation, our fundamental means of connection, and a language of expression. New algebraic systems will be constructed, using abstract objects like symmetries, sets, and functions themselves. Their structures will be investigated and compared with the structure of numeric operations like addition and multiplication, revealing the underlying essence of our number system. Extensive work will be done with functions and their roots in the Cartesian coordinate plane. New operations and inverses will lead to new functions, creating new equations, necessitating the inclusion of new numbers. This process of extending the field of numbers will culminate in the complex plane with the introduction of imaginary numbers. In addition, one-to-one functions will be used to extend the principles of counting to compare sizes of infinite sets.

ELECTIVES

TRIGONOMETRY

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 II.

TRIGONOMETRY/ANALYSIS

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. Text: Dolciani et al., Modern Introduction to Analysis. Prerequisite: Algebra II.

CALCULUS I

This is a college-level calculus course, with heavy emphasis on proofs, derivations, and creative applications. Limits, differentiation and integration, and applications thereof are covered. Transcendental functions are also explored. The course is intended for the serious mathematics student. Prerequisite: Trigonometry/Analysis.

TOPICS IN CALCULUS

This course is an inrtoduction to the basics of Calculus. It is designed to be an in-depth look at the fundamentals of the subject and will have less emphasis on some of the more challenging aspects. Thorughout the year, time will be taken to strengthen the algebraic skills necessary to solve Calculus problems, as well as to work on some of the more complicated skills in a supportive class setting. Students will begin with a look at the basic limit properties as well as some applications of continuity. The course will lead into a discussion of instantaneous rate of change, which will help develop a definition for a derivative. Students will explore the properties of derivatives and apply them to polynomials. Additionally, some time will be spent on trigonometric and exponential functions. The remainder of the year will be dedicated to the exploration of the area under a curve and development of the basics of integration.

SHAPE AND MOTION

Perfect mathematical objects like circles can exist only in our minds. What is the nature of this imaginary Platonic realm? What can be known and how can we know it? This course will trace the historical and philosophical development of the mathematics of measurement. Our intellectual journey will take us through classical geometry and trigonometry, the measurement of polygons and polyhedra, conics, and projective transformations, coordinate systems and vectors, mechanical curves, spacetime representations, and the differential and integral calculus. Our focus will be on the beauty and elegance of these ideas and their pivotal role in mathematical history.
This course will offer you the opportunity to engage in the actual practice of mathematics: posing and pursuing your own problems and conjectures, devising your own original arguments and explanations, collaborating as a mathematical community, and experiencing firsthand the joys and frustrations of creative mathematical work.

CALCULUS II

Calculus II is a continuation and expansion of the techniques of Calculus I. 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. Prerequisites: Calculus I and departmental recommendation.

ADVANCED PROBLEM SOLVING

This course is designed for students intending to participate in the school’s math team. We focus on mathematical topics not typically covered in the standard curriculum. Topics such as number theory and modular arithmetic, combinatorics, polynomials, geometric loci, probability, functional equations, algebraic and trigonometric identities, geometric inequalities, divisibility, colourings 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 independent research and present their discoveries at an in-school math fair. Prerequisites: Algebra II, 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’ll evaluate our games according to how well they model real-world scenarios, and how simple, fun, and enlightening they are to play. Note: This course can be taken for either a math or a science credit.

INDEPENDENT STUDY IN MATHEMATICS

Topics to be determined by interest and inclination of individual student and teacher.

MICROECONOMICS

This course is an introduction to the principles and applications of microeconomics. Topics to be covered include the theory of supply and demand, market equilibrium, consumer behavior, the behavior of firms, and perfect and imperfect competition. Social issues such as pollution, income distribution, and welfare are analyzed within an economic framework. Prerequisite: Algebra II.

MACROECONOMICS

This course is an introduction to the principles and applications of macroeconomics. Topics to be covered include the Keynesian and classical models of equilibrium, national income, inflation, unemployment, fiscal and monetary policy, investment and the banking system, international trade, and economic growth. Prerequisite:
Algebra II; Microeconomics is not a prerequisite for this course.

MATHEMATICS OF LIFE

We explore the use of mathematical models to understand biological processes. In the process, we investigate a diverse set of biological dynamics, including the genetic code, the relationship between structure and function of proteins, the forces that guide evolution in viruses, bacteria and eukaryotes, population dynamics, competition and cooperation among species, metabolism and catalysis, neural excitation and inhibition, immunological memory, origins and detection of life. The modelling 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. Our method of inquiry begins with gathering data and organizing our observations using graphs, and moves to conjecturing models of the apparent relationships, calibrating the models to the data and finally simulating the models computationally to make predictions, both quantitative and qualitative. The readings will include excerpts from E. Schroedinger, What is Life, F. Dyson, Origins of Life, S. Kaufmann, Origins of Order and At Home in the Universe, M. Eigen and R. Winkler, Laws of the Game, G. Rowe, Theoretical Models in Biology, and A. Wagner, Robustness and Evolvability in Living Systems. The class includes a weekly computer lab, culminating in a set of group projects, to be presented at the end of the year. Prerequisites: Biology and Algebra II, or permission of instructor. This course can be taken as either a math or a science credit.

NON-EUCLIDEAN GEOMETRY

The way that Euclid formalized his understanding of space held sway over the minds of human beings for over two thousand years. His system was hailed as certain knowledge by no less an intellect than that of Immanuel Kant. To contradict some part of Euclid’s system—especially something as intuitive as the way he treats parallel lines—surely this would lead to absurdity, or even madness! Our course will trace the historical thread of this system-questioning and the non-Euclidean geometries that it birthed, from the first ancient inklings to their explosion onto the mathematical scene in the nineteenth century. In the light of this historical context, the main thrust of the course will be a thorough exploration of the two classical non-Euclidean geometries: elliptic geometry (residence of the right equilateral triangle) and hyperbolic geometry (home to straight lines that approach each other, but that never meet). This will include writing proofs, solving problems, and building models. Of course, once one rule is broken, breaking others becomes less taboo, and so we will have the chance to look at geometries even further afield from the familiar Euclidean. Discussion of the implications of non-Euclidean geometries for philosophy and physics is certainly a part of this course. Prerequisite: Geometry

WHAT IS MATH?

What do mathematicians do, and why do they do it? This class examines the art of mathematics from both the philosophical and aesthetic points of view, providing a broad overview of the subject. Mathematics is about exploring our imaginations, finding beautiful patterns, and searching for explanations. Along the way we discover infinite numbers, the transcendence of pi, and the symmetry of knotted space. And we might just learn to see in four dimensions . . . The course features a survey of important unsolved problems that motivate modern research, as well as a “studio” where you create and critique your own works of mathematical art. The purpose of the course is to help you develop your mathematical intuition and taste, and in the process blow your mind to pieces. No previous mathematical experience is necessary, but permission of the instructor is required.

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