Description of Courses
The Basics: This is the most important math course the student will ever take. It lays the foundations for all subsequent math study. Understanding is built one axiom at a time. From an understanding of how numbers work, we meet fractions, exponents, decimals, and percents. One learns arithmetic from an algebraic point of view, and it makes sense, perhaps for the first time. Content includes material commonly encountered through grade 7, as well as problem solving techniques.
Middle-School Math: This is a combination of "Basics" and Algebra 1, designed to address the needs of middle-schoolers. We cover the content in Basics, and in Algebra 1 through the solution of first order equations. There is an emphasis on ego building.
Algebra 1: A continuation of Basics, described above; the work alternates among conceptual understanding , the skills of manipulation, graphing techniques, and the art of problem solving. Content includes the solution of first and second order equations, graphing and algebraic methods for the solution of systems, use of a calculator, and how to estimate results before ever touching a calculator.
Algebra 2: We return to first order equations for a bit of review—starting with “constant rate” word problems. Linear programming provides rich applications of graphing techniques and the solving of linear systems and for understanding inequalities. The solution of second order systems leads naturally to an exploration of the properties of functions and relations—exponential functions in particular. From there to the study of logarithms, their uses in modeling growth and decay, and applications in the worlds of science and commerce . These considerations lead to a need to understand sequences and series--- after which we are finally ready to fully understand the mathematical behavior of money; particularly why a sinking fund for a new car is a not only good idea, but just how good it is.
Statistics: We begin with probability, the basis of statistics. This leads to the notion of distributions. We learn ways to quantify an understanding of large quantities of numbers (measures of central tendency and dispersion), and see how very helpful a calculator is in this effort. From there to probability distributions and sampling distributions, where we meet the beautiful central limit theorem, then to point estimation techniques and hypothesis testing. We end with regression and correlation techniques.
Geometry: With this formal, axiomatic approach students learn the art of logical thinking while learning the usual plane geometry facts. The methods and principles used in determining the validity of arguments are introduced, conditional as well as indirect. Proof is emphasized throughout the course. Students love this course—they begin to feel powerful.
Trigonometry: In this subject the student meets substantive real world situations. It is here that problem solving skills are sharpened and algebra skills are solidified. This is a classical, in depth trig course. Material will include a visual presentation of what the trig functions are, where their names came from, their graphs and why those are useful, , identities and their use in solving trig equations, the complex plane and a study of vectors—an ease with which is indispensable for the study of physics.
Pre-Calculus: In this course the student “meets it all again”, but this time the emphasis is on function theory. Content includes exponential growth and decay, harmonic motion, and the math needed to understand , describe and make predictions under those circumstances, techniques for solving higher order polynomial equations, analysis of the conic sections, and further study of the behavior of sequences and series--mainly the nature of convergence .
AP Calculus AB: Calculus I: In addition to the traditional analytic approach to calculus, we present a visual demonstration of the connection between the “rate of change” problem and the “area” problem, as well as the “change in area” problem--the ideas embodied in the two Fundamental Theorems of calculus. Our objective is to see that the student has more than just a mechanical understanding of the basic ideas of the calculus, because this is what allows the student to "own" the material--to be able to use it at will. There is a heavy emphasis on problem solving—problems in all sorts of areas: business, sociology, medicine, physics, astronomy. This course will prepare the student to pass the national AP Exam in Calculus AB given in early May. Content includes limits, derivatives, inverse functions (logarithmic, hyperbolic and trigonometric) and their derivatives, indeterminate forms, the Mean Value Theorem, curve sketching, the integral, and applications of integration and techniques of integration. There is a strong emphasis on problem solving techniques and on practical applications.
AP Calculus BC: Calculus II: Subject matter includes the content of Calculus I, above, plus parametric equations, arc lengths and surface areas, polar coordinates and conics; infinite series and sequences, especially Taylor Polynomials; three-dimensional analytic geometry, cylindrical coordinates and vector algebra with applications to quadratic surfaces and motion in space.
Calculus III: This course completes the undergraduate study of calculus. Curriculum includes partial derivatives, Legrange multipliers, multiple integrals and their applications, vector calculus including Stokes’ and Green’s Theorems, and, time permitting, and introduction to differential equations.
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