Mathematics

Many of our students are destined to be leaders in fields where analytical and problem-solving skills are invaluable, and specific applications of mathematics are often indispensable. In the Mathematics Department, we give each student the necessary tools to understand theories and appreciate applications. True problem-solving strength calls for a wide repertoire of knowledge. Students acquire a strong knowledge of mathematics through the study of algebra, geometry, trigonometry, calculus, and statistics. This ability to solve problems is further strengthened by learning to discern relationships between particular skills and concepts and the fundamental principles that unify them. A statistics curriculum is incorporated into math courses at all levels, not only preparing students for further studies of statistics in their senior year, but giving them the tools needed for research in upper division science courses and the Marine Ecology Research program. At all levels of instruction, students are comfortably challenged as they learn to formulate key questions, collect and analyze data, and apply learned strategies to new situations. In doing so, students are equipped not only to solve mathematics problems but also to use an analytical, well-thought-out approach in seeking solutions in other areas of life.

Algebra 1

This class introduces the basic principles necessary for success in future mathematics courses. Students transition from the concrete to the abstract through a wide range of problem-solving situations. The class emphasizes the concept of functions and covers the real number system, operations with positive and negative numbers, simplifying algebraic expressions, solving and graphing linear equations and inequalities, applying rules of exponents, understanding operations involving polynomials, simplifying rational expressions and square roots, solving systems of linear equations, and solving both rational and quadratic equations.

Algebra 2

This course aims for a mastery of the manipulative skills in real-number algebra as well as further developed problem-solving skills. Students review and extend their ability to manipulate polynomial and rational expressions and to solve linear, quadratic, fractional, and radical equations and inequalities. The course includes the study of irrational and complex numbers, matrices, conic sections, nonlinear systems of equations, sequences, series, right triangle trigonometry, and concludes with an introduction to exponential and logarithmic functions.

Algebra 2 Honors

This course aims for a mastery of the manipulative skills in real-number algebra, and the pace is set to challenge students with exceptional mathematical abilities. Students review and extend their ability to manipulate polynomial and rational expressions and to solve linear, quadratic, fractional, and radical equations and inequalities. New topics include systems of linear equations and matrices, rational exponents, sequences and series, as well as higher-degree polynomial, inverse, exponential, and logarithmic functions. The course concludes with instruction in trigonometric functions, equations, identities, and graphs.

Geometry

This course introduces students to concepts in geometry and teaches how to write two-column and paragraph proofs. Basic algebraic skills are reinforced throughout this course with special emphasis on applications of congruence and similarity of triangles and other polygons, properties of circles, right triangles, coordinate geometry, transformations, areas of plane figures, and volumes of solids. Throughout the course, students will carry out geometric constructions using a compass and straightedge, as well as Geometer’s Sketchpad. The course concludes with instruction in trigonometric functions, equations, identities, and graphs.

Geometry Honors

This course is for students who have a genuine interest and high aptitude in math. In the class, students gain a greater appreciation of the nature of a mathematical system through the study of challenging mathematical proofs and the application of advanced algebra skills. Other topics emphasized are applications of congruence and similarity of polygons, properties of circles, transformations, coordinate geometry, analytic geometry, areas of plane figures, and volumes of solids. Also, inductive and deductive reasoning are discussed, and constructions are completed using a compass and straightedge, as well as Geometer’s Sketchpad.

Precalculus

This course is an introduction to analysis. The intent is to utilize all the mathematical concepts developed in previous math courses and to sum up the basic concepts of mathematics. In preparation for college-level calculus, precalculus integrates a number of topics, including functions, inverse functions, theory of logarithms, functional trigonometry, polynomial equations, probability, statistics, combinatorics, limits, continuity, and the definition of the derivative.

Precalculus Honors

This course offers a challenging introduction to the study of mathematical analysis. To successfully manage the pace and challenging content of this course, students must be able to retain and apply mathematical knowledge, as well as persevere when solving unexpected problems. This course integrates a number of topics, including an in-depth study of functions and their graphs, theory of logarithms, trigonometry, polynomial equations, combinatorics, probability, and statistics. Students are also introduced to limits, continuity, and the definition of the derivative.

Advanced Placement Calculus AB

This course is roughly equivalent to a first semester college calculus course devoted to topics in differential and integral calculus. The Advanced Placement course covers topics in these areas, including concepts and skills of limits, continuity, derivatives, definite integrals, the Fundamental Theorem of Calculus, and applications of differential and integral calculus. Students who are considering college majors such as business, engineering, mathematics, science, and economics should consider taking this course in preparation for further college calculus requirements. Upon completion of this course, students will sit for the Advanced Placement exam in May.

Advanced Placement Calculus BC

This course is roughly equivalent to both first and second semester college calculus courses. It extends the content learned in Advanced Placement Calculus AB to different types of equations and introduces the topics of infinite series, limits of indeterminate forms, improper integrals, and parametric, vector, and polar functions. Students who are considering college majors such as business, engineering, mathematics, science, and economics should consider taking this course in preparation for further college calculus requirements. Upon completion of this course, students will sit for the Advanced Placement exam in May.

Statistics

This introductory, non-calculus-based statistics course will introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: 1) exploring data: describing patterns and departures from patterns; 2) sampling and experimentation: planning and conducting a study; 3) anticipating patterns: exploring random phenomena using probability and simulation; and 4) statistical inference: estimating population parameters and testing hypotheses. Students who are considering college majors such as social science, health science, business, engineering, mathematics, and science should consider taking this course in preparation for further college statistics requirements.

Math 4: The Nature of Math & Personal Finance

This course provides students with the opportunity to broaden their mathematical experience and to understand how mathematical thinking is useful in their daily lives. During the fall semester, students will have the opportunity to explore the concept of infinity, discover how patterns can be used to solve problems involving a variety of mathematical topics, and fold paper to create various geometric solids. During the spring semester, students will have the opportunity to learn about essential elements of personal finance that they are likely to encounter as young adults both during and after college. Students will learn about interest, the present and future value of money, debt, basic banking, investing, loans, retirement savings, insurance, and taxes.

Introduction to Programming

The purpose of this one-semester class is to introduce students to the concept of programming; no prior experience with computers and computer programming in particular is assumed. Students learn through short lectures, individual and group work during class, and by completing programming assignments and projects of varied complexity. Assignments are tailored to challenge interested, highly motivated students, and yet be accessible and interesting to students requiring more individual guidance. The course is based on the Processing programming language, a Java-derived language. Processing allows students who wish to continue their adventure with programming a relatively simple transfer to Java and JavaScript. At the same time, Processing, which is often referred to as the programming language for artists, makes the first steps in programming particularly rewarding since it allows for the creation of visually appealing graphical projects with very limited programming background needed.

Computer Science and Technology

This full-year, rigorous course introduces students to the foundations of modern computing. The course covers a broad range of topics such as programming, algorithms, the Internet, big data, digital privacy and security, and the societal impacts of computing design. The course seeks to provide students with a foundation in computing principles so that they are adequately prepared with knowledge and skills to meaningfully participate in our increasingly digital society, economy, and culture. There are no prerequisites for this course. However, the majority of students will have taken the Introduction to Programming course at Santa Catalina. Thus, we continue to build on programming skills using the Processing language, Java, and JavaScript.