Mathematics

Algebra I focuses on the study of elementary functions, including linear equations, linear systems, and quadratic equations. The course also includes work on polynomials, rational algebraic expressions, laws of exponents, and simplifying radicals. An introduction to the use of the TI Nspire graphing calculator app (a graphing calculator) is also part of the course. A major emphasis of the course will be the ways that mathematics is used to solve “real world” problems. Students will also have the opportunity to apply their skills in a variety of mathematics projects throughout the year. (This course will be different in structure from the two-year course offered in Middle School; however, all major first year algebra topics will be covered.) 

This course includes an in-depth analysis of plane, solid, and coordinate geometry as they relate to both abstract mathematical concepts as well as real-world problem situations. Topics include logic and proof, parallel lines and polygons, perimeter and area analysis, volume and surface area analysis, similarity and congruence, trigonometry, and analytic geometry. Emphasis will be placed on developing critical thinking skills as they relate to logical reasoning and argument. Prerequisite: Algebra I 

Geometry is the study and measurement of shapes, and its study is both extremely useful and very beautiful. Moreover, we discover geometric truths by carefully applying the rules of logic, which are valuable in their own right throughout our lives. In Geometry Advanced, we explore directly the rules of symbolic logic and Boolean Algebra, distinguishing assumptions, definitions and proven statements. Using these tools, we consider points, lines and planes and apply these notions to in-depth studies of two-dimensional shapes – triangles, more complex polygons, and circles and three-dimensional shapes – polyhedrons and spheres – while integrating concepts from algebra at every step to establish the foundations of analytic geometry. After comparing the notions of congruence and similarity, we go on to develop the foundations of trigonometry. Throughout our work, we study both theory and real-word applications, and we discuss the historical and cultural context in which geometry developed and has been used over the centuries. These studies form the theoretical and practical foundation for all the advanced studies in our Upper School curriculum, including Algebra II, Precalculus, Calculus, Statistics and Linear Algebra. Prerequisite: Algebra I and recommendation of the department 

In the first part of this course, students study the properties that make a mathematical relationship a function. They learn how various elements of the function equation determine transformations of the graph of the specific function. The functions that are studied are linear, quadratic, rational, radical, exponential and logarithmic functions. In a world that is data driven, it is important that students be introduced to the topics of matrices, data analysis, and probability. Each of these topics introduces the student to a way of organizing and analyzing data that will help them in a decision-making process. Prerequisite: Algebra I

The difference between our advanced and regular sections is in the speed at which information is introduced and the difficulty of the problems required of the students. In addition, the advanced class studies two units not included in our regular Algebra II curriculum. 

In the first part of the course, students study the properties that make a mathematical relationship a function. They learn how various elements of the function equation determine transformations of the graph of the specific function. The functions that are studied are linear, quadratic,rational, radical, exponential and logarithmic functions. In a world that is data driven, it is important that students be introduced to the topics of matrices, data analysis, and probability. Each of these topics introduces the students to a way of organizing and analyzing data that will help them in a decision making process. In addition to the above topics which are also covered in our Algebra II regular course at a more elementary level, this class includes an introduction to the study of sequences and series which is necessary for success in our Calculus Advanced classes. Prerequisite: Algebra I and recommendation of the department

This course is designed to advance the student’s understanding of mathematical functions, trigonometry, and modeling and give students an introduction to data analysis and statistics. Students are asked to investigate functions, their graphs, and their applications. The course strives to help students understand the connections between the representations of functions graphically, numerically, and analytically. Students also develop mathematical models for various “real world” situations. In this course, more of an emphasis is placed on applications than on theory. Students will also make extensive use of the Desmos online graphing calculator. Prerequisite: Geometry and Algebra II 

Students in Calculus I work in groups to explore conceptual calculus concepts. Instead of a textbook, the CalcMedic curriculum has been adapted for this course. Students explore concepts together in groups first, and then we formalize them together as a class. This course covers differential calculus and an introduction to integral calculus. For differential calculus, the importance of the connection of instantaneous rate of change to the slope of tangent lines to a curve is emphasized as a way of thinking about the first derivative. For integral calculus, the visual basis of the understanding of area under a curve is the accumulation of an increasing number of rectangles determined by the curve. We will cover the basic derivative rules and integration rules for the functions that were studied in Precalculus. Prerequisite: Precalculus or Advanced Precalculus 

Students in Precalculus Advanced explore challenging problems and concepts throughout the curriculum. They work in groups and independently and use the desmos graphing calculator to explore new functions and to ask new questions. The course’s main focus is an intensive look at mathematical functions in all their various forms. This is a significantly more conceptual course than Precalculus. Topics include polynomial, rational, exponential, logarithmic, and trigonometric functions, as well as their transformations and inverses. Time permitting, additional topics may be included such as probability or conic sections. Prerequisite: Geometry and Algebra II Advanced and recommendation of the department. There is a placement test for this course.

This course is a more rigorous development of single variable differential and integral calculus from multiple perspectives (algebraic, graphical, numerical, and verbal). For differential calculus, the importance of the connection of instantaneous rate of change to the slope of tangent lines to a curve is emphasized as a way of thinking about the first derivative. We will look at the application of the derivative to solve related rate questions, optimization questions and the application of derivatives to the graphing of functions. For integral calculus, the visual basis of the understanding of area under a curve is the accumulation of an increasing number of rectangles whose height is determined by the curve. We will use the definite integral to find the area under a curve or between curves, volumes of revolution around an axis using the disk method and the shell method. We end the course looking at integration by parts. We will cover the basic derivative rules and integration rules for the functions that were studied in Precalculus. Prerequisite: Precalculus Advanced 

This course is for students who have excelled in Calculus I Advanced. The Calculus II Advanced curriculum is even more challenging and typically includes advanced techniques of integration, calculus in polar coordinates, graphing, infinite series, first-order ordinary differential equations and applications, and three-dimensional coordinate systems. In addition to a deep understanding and retention of material covered in Precalculus Advanced and Calculus I Advanced, successful Calculus II Advanced students will have demonstrated both successful completion of assigned work and an intuitive grasp of the subject matter.  Prerequisite: Calculus I Advanced 

This course provides a foundation for future statistics courses and develops the student’s ability to understand data in the world at large. Students develop their ability to gather, organize, interpret, and present data, as well as learn how to use the basic tools of probability. These topics are presented through activities, projects and group problem solving. We use Desmos, applets, and other statistical software programs as computational and analytical tools In this course, as opposed to Statistics Advanced, there is additional focus on exploring data, planning a statistical study, and anticipating patterns using probability and the unit on confirm ing models through statistical inference is deemphasized. Prerequisite: Algebra II 

This course is equivalent to an introductory, non-calculus based college course in statistics. This accelerated course is intended to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad themes: 1. Exploratory data analysis, such as using graphical and numerical techniques to study patterns and departures from patterns. 2. Data collection and planning a statistical study so that valid data are obtained. 3. Anticipating patterns using probability, and 4. Confirming models through statistical inference. In and out of class students often work in groups to complete activities and projects. The course makes significant use of the R programming language, applets, and calculators to perform most of the statistical calculations, leaving the analysis and interpretation of the results to the students. Students who successfully complete this course have the option of taking the Advanced Placement test in Statistics and are adequately prepared for the AP Exam. Prerequisite: Precalculus

The Linear Algebra course offered at Friends’ Central School is comparable to a college-level first course in Linear Algebra for math, engineering, and computer science majors and is an excellent choice for students who plan to pursue these disciplines in college and beyond. Points of emphasis include matrices, vector operations, vector space basics, linear dependence/independence, basis, span, and invertibility. Students are introduced to basic mathematical proof, a small but important component of the course, and are able to shape their own Linear Algebra experience through selection of projects and applications. There is, indeed, more to life – and math! – than Calculus, and we are pleased to offer this relevant and challenging option to our upper-level math students. Prerequisite: Calculus II Advanced, a willingness to work independently, and an interest in mathematics and abstract thinking 

Economics is the science of decision making, the study of choice - where the costs and benefits of alternatives are evaluated. This semester-long course will be an exploration of certain concepts in economics. The main goal of the course is two fold: 1. to understand basic economic concepts and 2. strengthen critical thinking skills needed to understand and respond to various complex situations. The course will require utilization of various mathematical skills to comprehend the economic concepts presented. The course will be split into three units: Microeconomics, Macroeconomics, and Business Ethics. Upon completion, students will understand fundamental economic concepts, appreciate how the principal concepts of economics relate to each other, and understand the structure of economic systems. Students will be able to use economic concepts in a reasoned, careful manner in dealing with personal, community, national, and global economic issues. Prerequisite: Precalculus