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, and calculus. 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. Students learn to formulate key questions, analyze data, and apply learned strategies to new situations. In doing so, they 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 of 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. Mastery of these topics are required for success in Algebra 2.

Geometry

This course introduces students to concepts in planar and three-dimensional geometry, and teaches students how to write algebraic, two-column, and paragraph proofs. Important algebraic content from Algebra 1 is reinforced throughout this course, and Geometer’s Sketchpad and GeoGebra are used to help students visualize geometric concepts and examine various geometric relationships in a dynamic environment. Topics covered in this course are geometric probability, congruency, similarity, polygons, coordinate geometry, properties of circles, areas of plane figures, surface area and volumes of solids, geometric transformations, constructions, and trigonometry.

Geometry Honors

Students who have a keen interest and high aptitude in math are best suited for this class. Students gain a greater appreciation of the nature of a mathematical system through the study of challenging mathematical proofs and constructions. Other topics emphasized are applications of congruence and similarity of triangles, properties of circles, polygons, inductive and deductive reasoning, coordinate geometry, areas of plane figures, surface area and volumes of solids, geometric probability, transformations, and trigonometry. Also, important algebraic content from Algebra 1 is reinforced throughout this course.

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 exponential and logarithmic functions.

Algebra 2 Honors

This course emphasizes creative problem-solving and mastery of a variety of mathematical functions. The pace is set to challenge students with high mathematical abilities. This advanced algebra class begins with a brief review of important concepts and procedures learned in Algebra 1 and proceeds into greater complexities, with subsequent introduction of the elements of exponents, functions, polynomials of higher degrees, logarithms, sequences, series, systems of linear and nonlinear equations, matrices, probability, statistics, conics, and trigonometry.

The Nature of Math

This semester course gives students the opportunity to explore the concept of infinity; to study mathematical patterns in nature, art, and music; to create various geometric solids; and to discover how patterns can be used to solve problems involving geometry, numeration systems, networks, topology, exponential growth and decay, and fractals. As students build skills in problem-solving and estimation, they also gain a better understanding of the historical development of mathematical ideas.

Personal Finance

This semester course gives students 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 learn about interest, the present and future value of money, debt, basic banking, investing, loans, retirement savings, insurance, and taxes. Throughout this course, students explore the nature of growth and decay, and compound interest. Overall, the course focuses on solving real-world problems and providing students with the basic knowledge and tools they will need to apply their problem-solving abilities to their financial life.

Pre-calculus

This course is an introduction to mathematical analysis. Students continue to study functions and trigonometry, both graphically and analytically, along with topics in discrete mathematics. The intent is for students to utilize all of the mathematical concepts and procedures developed in previous math courses so they can access and apply this knowledge effectively when solving calculus problems the following year. The course concludes with a study of limits, continuity, the definition of the derivative, and basic differentiation techniques.

Pre-calculus Honors

This course offers an extensive, yet swift, review of functions both graphically and analytically. Further studies involve analytical trigonometry, applications of trigonometry, statistics, and discrete mathematics, as well as a challenging introduction to calculus. Much of the second semester is devoted to learning calculus concepts and procedures related to limits, continuity, differentiation, and integration. Students are expected to master the course content and demonstrate an ability to apply mathematical knowledge efficiently and effectively when problem-solving.

AP Calculus AB

This course offers students the opportunity to take a college-level mathematics course in preparation for more advanced studies of calculus. Course content includes the list of topics set forth for Calculus AB in the College Board’s Course Exam and Description (CED) Book and more. Students study differential and integral calculus in a variety of ways so that they are able to approach and solve problems involving calculus numerically, analytically, graphically, and verbally. Throughout the course, students are gradually introduced to AP-type questions and are expected to solve a number of multiple-choice and free-response questions from past exams with and without the use of a graphing calculator. Problem solving in class provides students the opportunity to engage with one another so they can communicate their solutions clearly in both verbal and written forms. AP exam preparation is emphasized during the final three to four weeks of the course.

AP Calculus BC

This course offers students the opportunity to take a college-level mathematics course in preparation for more advanced studies of calculus or other advanced mathematics topics. Course content includes the list of topics set forth for Calculus BC in the College Board’s Course Exam and Description (CED) Book and more. The depth and pace of this course is appropriate for talented and curious math students with tremendous self-motivation and discipline. Students study the course content in a variety of ways so that they are able to approach and solve problems involving calculus numerically, analytically, graphically, and verbally. Throughout the course, students are gradually introduced to AP-type questions and are expected to solve a number of multiple-choice and free-response questions from past exams with and without the use of a graphing calculator. Problem-solving in class provides students the opportunity to engage with one another so they can communicate their solutions clearly in both verbal and written forms. AP exam preparation is emphasized during the final three to four weeks of the course.

Advanced Topics: Statistics

Statistics introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students develop statistical strategies from a wide variety of sources including experiments, sample surveys, and other observational studies. Students study probability and simulation to aid in their understanding of statistics and to aid in constructing models of chance. Also, students study the properties of sampling distributions in order to construct confidence intervals and to carry out a variety of statistical tests. Throughout the course, students use technological tools such as graphing calculators and spreadsheets to organize, view, and analyze data.

Advanced Topics: Multivariable Calculus

This yearlong course explores topics that are studied in a typical third semester college-level calculus course. It begins by reviewing key topics from the AP Calculus BC curriculum, such as parametric equations, the polar coordinate system, and vectors. Students are introduced to vector-valued functions, and the geometry of space are taught, as well as partial derivatives and multiple integrals. Vector calculus topics taught include vector fields, line integrals, Green’s Theorem, surface integrals, Stokes’ Theorem, and the Divergence Theorem. The course concludes with an introduction to elementary differential equations.

Introduction to Programming

The purpose of this semester-long class is to introduce students to the concept of programming in a fun and engaging manner; no prior experience with computer programming 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 taught using procedural programming and object oriented programming in the Python language. Python makes the first steps in programming particularly rewarding since it allows for creativity with visually appealing graphical projects. Graphics also provide visual cues when a logical solution has errors, which aids in debugging.

Advanced Topics: Computer Science & Technology

This yearlong course covers a broad range of topics including object-oriented programming, algorithm development, physical computing, networks, and virtual reality three-dimensional design. The course also seeks to provide students with a foundation in computing principles so 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 completed the Introduction to Programming course at Santa Catalina. Thus, this class continues to build on the programming skills using the Processing language as well as other languages, including Arduino, HTML, and JavaScript. Students demonstrate their skills and understanding through regular nightly homework, presentations, long-term projects, quizzes, and exams.