## Math Courses

Advanced Geometry is a one semester, upper level elective course that pushes students to express geometric and related mathematical ideas through visual and tactile means. Possible topics explored in depth include fractal art, tessellations and Islamic tile art, topology, map making, origami, knots & weaving, Bezier’s curves, optical illusions and anamorphic art, and non-Euclidean Geometry. Student-driven projects will be the primary means of demonstrating knowledge and success in the class, and will heavily depend on computer based drawing and modeling applications such as Geometer’s Sketchpad, GeoGebra and Google SketchUp. Three dimensional art and construction projects will complement the digital learning environment.

This course is open to those who have completed Algebra II.

This course will be a full year course in conjunction with (Matrix Algebra or History of Math).

In the year long course, Advanced Topics in Calculus (ATC) we will be studying many advanced topics. These may include: more advanced differential equations, parametric equations, polar equations, vectors, more advanced applications of derivatives and antiderivatives, advanced matrix algebra, more advanced series and sequences, and more that will be decided amongst the class.

This course is open to those who have completed Calculus.

Algebra is the foundational skill for manipulating quantitative relationships involving equality, inequality and correlation. It is a symbolic language that also allows one to solve problems from the physical, natural, and business worlds. Students strengthen their understanding of Algebra when they continuously make this connection between abstraction and real-world contexts through their use of equations, tables, graphs, diagrams and narratives. The primary topics in this course include: terminology and properties of the set of real numbers; creating, simplifying and translating algebraic expressions; writing, solving and graphing linear equations (including proportions and inequalities); the rules and properties of linear functions, systems of linear equations; performing operations with polynomials, especially factoring; solving absolute value equations; and an introduction to solving and graphing quadratic functions. Students are assessed primarily through lectures, nightly homework assignments, and quizzes and tests. At Darrow, Algebra I is the first in the series of three required year long courses, typically followed by Geometry and then Algebra II.

Algebra II is distinguished from Algebra I in two ways. First, it emphasizes an increased level of proficiency in the fundamental skill set of manipulating variables and equations, such as graphing, factoring and simplifying. Second, it stresses the central importance of functions in understanding any quantitative relationship. Each new unit covers a different “family” of functions and explores the overlapping ways of describing their properties, graphing their key points and end behavior, solving their roots and modeling their applications in real-world phenomena. The primary topics in this course include: a review of linear functions and inequalities; solving systems of linear equations graphically, algebraically, and with matrices; quadratic functions, higher degree polynomial functions, and rational exponent and radical functions. Depending on the class, time is spent on exponential and logarithmic functions, as well as an introduction to the unit circle and trigonometric functions. In each unit, the TI-84 graphing calculator is used extensively, both as a utility for calculations as well as a tool for exploring function properties. At Darrow, Algebra II is the last in the series of three required year long courses, typically preceded by Algebra I and then Geometry. Most students continue on to our higher level elective courses, depending on the year they complete Algebra II, as well as their post-high school goals and interests. Students who successfully complete Algebra II are well-prepared for the mathematics portion of the SAT.

Calculus is an advanced mathematics topic that requires abstract thought. The first semester is devoted to the derivative as defined by the slope of a curve; students begin by investigating limits and use this concept through formal proofs to define derivative. As the semester continues, students look at increasingly complex ways in which to take derivatives of various common functions. During the second semester, students investigate the integral, as defined by area under a curve. This study begins with a look at Riemann sums and antiderivatives, and progresses to more complex ways in which to take integrals, including substitution, integration by parts, algebraic identities, and improper integrals. The second semester ends with the study of practical applications of the integral.

Prerequisites: Pre-Calculus or permission of Math Department Chair.

Geometry is essential as a means to giving students the quantitative skills to describe and measure the 2D and 3D world. It is a fundamental part of understanding such diverse applications as visual arts, navigation, construction, design and architecture, etc. It is also the classic manner of fostering the mathematical skill of logical reasoning and proof. And it provides a visual means of reinforcing and enhancing the algebraic skills that were stressed in Algebra I and will be expanded upon in Algebra II. Thus, at Darrow, Geometry is the second in the series of three required year long courses. The primary topics in this course include: a review of the properties of points, lines, angles & polygons, an introduction to logical statements and proof, parallel and perpendicular lines, triangle congruence, ratios and proportions, right triangle trigonometry, properties of quadrilaterals, properties of circles, arcs and chords, two dimensional measurement involving perimeter, circumference and area, properties of three dimensional solids, as well as measurement of these solids involving surface area and volume. In addition to lectures, nightly homework assignments, quizzes and tests, students are assessed through lab exercises involving Geometer’s Sketchpad, a computer drawing program that can be used both analytically and creatively.

This course will discuss the history of mathematics, emphasizing the contributions of outstanding persons and civilizations (from Ancient Babylonia and Egypt through Greece,the Middle East, Asia, and on to modern Europe and USA). We will be seeing how the math develops from Babylonian tablets of Pythagorean triples to Andrew Wiles’ proof of Fermat’s last theorem. This course will have mathematical ideas from many different disciplines (algebra, geometry, probability, number theory, proofs, and calculus: geared to the level of mathematical ability of students enrolled). This course will have short essays as well.

This course is open to those who have completed Algebra II.

This course will be a full year course in conjunction with Advanced Geometry and Design.

Matrix Algebra is an in-depth study of matrices and their properties. Course topics include, but are not limited to, adding, subtracting, and multiplying matrices, an introduction to vector algebra, solving matrix equations through inverse and Gauss Elimination methods, linear dependence and independence, linear transformations, subspaces, and calculating eigenvalues and eigenvectors. If time permits, the class will move on to cover powers of matrices, the Gram-Schmidt process, and/or QR factorization.

This course is open to those who have completed Algebra II.

This course will be a full year course in conjunction with Advanced Geometry and Design.

Pre-Calculus is an in-depth study of functions and ways in which they can be manipulated. Course topics include, but are not limited to, combinations and composition of functions, graphing transformations, exponential and logarithmic functions, trigonometry, rational functions, conic sections and an introduction to limits. Pre-Calculus prepares students for Calculus by providing them with greater understanding of fundamental concepts of Algebra.

Prerequisites: Algebra II