The MCDS math program has two clear goals: to allow students a full range of choices in college, and to develop their intrinsic appreciation and profound understanding of the power and beauty of mathematics. Students learn the fundamental algebraic, trigonometric and calculus skills necessary for further mathematical study. Students also develop an expertise in the subject, enabling them to experience the beauty of higher-order math.

9th Grade Algebra and Geometry
Like all mathematics at MCDS, this course emphasizes the ability to “see” mathematics both algebraically and geometrically. Square roots introduce the irrational numbers; we prove the square root of two is irrational, compute rational approximations, and work with radicals. Students factor polynomials. They solve quadratic equations by factoring, by completing the square, and by the quadratic formula, which we derive. We build general understanding of functions by working with specific functions. Geometry includes the circle, similarity of plane figures and of solids, and the theorem of Pythagoras which students apply to plane figures and solids. Some combinatorics, probability, and statistics are included. Through lecture, discussion and purposeful practice, students in this course develop the knowledge, skills, intuitive sense and confidence necessary for continued success in mathematics.

10th Grade Algebra, Coordinate Geometry, and Trigonometry
Numbers are extended to the reals, whose correspondence with the points of a line is emphasized, then to the complex numbers, enabling students to find all roots of any quadratic equation. We apply the remainder and factor theorems to higher degree polynomials. We define integral and rational exponents, practice the exponent rules, and simplify rational expressions involving polynomials. Students solve systems of linear equations, and systems of a linear and quadratic equation. We study polynomial, rational, and inverse functions. Coordinate geometry encompasses the straight line, the circle, distance, midpoint, parallel and perpendicular lines, intersection, and tangency of a circle and line. Trigonometry begins with trigonometric ratios, is generalized by the unit circle, and includes values of the functions at common angles, the law of sines, the law of cosines, the area of a triangle, and Heron’s formula. We stress the importance of definitions and statements of mathematical results and their proofs and converses. Students write direct and indirect proofs. We solve linear and quadratic inequalities. Combinatorics includes permutations, combinations, and Pascal’s triangle.

11th Quadratic Curves, Vector Geometry, Linear Algebra, and Basic Analysis
Students study conics. We introduce vectors of two and three dimensions, discussing components, magnitude, addition, subtraction, scalar multiplication, inner product, special vectors, parametric equations, vector equations of a straight line, circle, plane, and sphere. We study matrices including addition, subtraction, scalar and matrix multiplication, and inverse matrices. We discuss linear transformations, linearity, composition, mapping, and inverse mapping. Basic analysis introduces exponential and logarithmic functions, and deepens understanding of trigonometric functions including the addition formulae. Students work with arithmetic and geometric sequences and series. They write proofs using mathematical induction. Discussion of the binomial theorem draws on combinatorics of prior years. Fundamental ideas of calculus include limit, derivative, and indefinite and definite integrals. Eleventh grade analysis seeks to instill an intuitive sense of several ideas that underlie the calculus. The treatment is informal.

12th Grade Calculus
This college level course covers single variable calculus using a college textbook. We treat limit and other fundamental ideas and theorems in depth. Students study algebraic, trigonometric, exponential, and logarithmic functions throughout the year. Differentiation is introduced through the tangent problem; the definite integral through the area problem. Throughout the course, geometry is emphasized. Techniques of integration include substitutions, integration by parts, and use of trigonometric identities and partial fractions. We consider indeterminate forms and improper integrals. We introduce polar coordinates, complex numbers in polar form, and De Moivre’s theorem. Students working with sequences and series learn the basis for, and application of, tests for convergence. We discuss power series, Taylor and Maclaurin series, and Newton’s method. Students use a function’s derivatives to understand its behavior. Additional applications of the derivative are taken from the sciences. Using integration, students find the area of a plane region, volumes of solids and of solids of revolution, the length of a plane curve, moments, and center of mass.

Students will not be rushed through the material, but will be given time, problems and discussion sufficient for them to develop the sense, insight and formal skill that constitute true ability to do and use calculus. Limits will be treated in some depth.