"We prepare students to engage in the world that is and to help bring about a world that ought to be."

Course Catalogue

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Mathematics

Mathematics is a universal language whose symbolic logic and precision of thought is shared by all cultures.  Mathematics allows us to measure, quantify, organize, analyze and visualize relationships in the natural world, in the sciences, in society, and among abstract concepts. From the fundamentals of counting to the study of rates of change in calculus, we strive to teach our students not only to calculate, but also to appreciate the inherent beauty, power, and elegance of mathematical patterns and processes.  Mathematical truth can be seen as reflecting the Quaker values of simplicity and integrity. The discipline of mathematics fosters academic focus, concentration, and perseverance in problem solving that can be applied to all a student’s studies.

While in the Upper School, each student is required to take the equivalent of three years of math courses, including Geometry and Algebra II.  We attempt to strike a balance between ability grouping and open-ended placement, evaluating each student’s performance, goals, and readiness each year.  After Algebra II, students place into a Precalculus course and may qualify to take a Statistics course concurrently. Following the Precalculus level, students may take a Calculus course, a Statistics course, or a different math elective. Multivariable Calculus is offered to students who take Advanced Calculus I & II prior to their senior year.  If a student wishes to co-enroll in two math courses, they must schedule a meeting with the department chair to discuss additional requirements for co-enrolling.
 

  • Geometry with Algebra Applications

    This course introduces geometric concepts through an inductive approach, observing patterns and drawing conclusions. Students build on previous algebraic skills to strengthen their conceptual understanding of mathematics and apply them to geometrical concepts. Students formulate definitions and conjectures involving polygons, circles, three-dimensional figures, parallel and perpendicular lines, congruence and similarity, area and volume, the Pythagorean Theorem, coordinate geometry, and right triangle trigonometry. Students also have the opportunity to explore the deductive structure of mathematics and apply mathematical concepts to the physical world. 
    5 periods/cycle   
    Full-year course – 4 credits
    Prerequisite(s): Algebra I or the equivalent and permission of the department   

  • Geometry: Proof-Based

    This course explores the nature of spatial relationships and the properties of certain geometric figures while helping students understand the nature of an axiomatic deductive system.  Students study the role of postulates, definitions, and theorems, analyze classical proofs, and learn how to write original proofs. Geometric topics studied include: triangles, quadrilaterals, polygons, circles, basic three dimensional shapes, parallel and perpendicular lines, congruence and similarity, area and volume, the Pythagorean Theorem, and an introduction to right triangle trigonometry. Continued algebra review is integrated into various geometric concepts.  Applications to the physical world are included where appropriate.

    5 periods/cycle   
    Full year course – 4 credits
    Prerequisite(s): Algebra I or the equivalent and permission of the department                                                                                                              

  • Algebra II/Trigonometry

    This course applies algebra skills to an increasingly integrated abstract level. Topics include linear and quadratic functions, absolute value functions, logarithmic and exponential functions, rational and radical functions, trigonometric functions and probability. For each function type, students simplify expressions, solve equations and graph functions.  Connections between algebraic and graphical representations of functions are emphasized, and graphing calculators are used throughout the course.

    5 periods/cycle   
    Full year course – 4 credits
    Prerequisite(s): Geometry: Proof-Based or Geometry with Algebra Applications  or the equivalent and permission of the department          
                                                                                                        

  • Algebra II/Trigonometry A

    This course applies algebra skills to an increasingly integrated abstract level. Topics include linear and quadratic functions, absolute value functions, logarithmic and exponential functions, rational and radical functions, trigonometric functions, and probability. For each function type, students simplify expressions, solve equations, and graph functions. Connections between algebraic and graphical representations of functions are emphasized and graphing calculators are used throughout the course. Students in this course cover the same topics as in Algebra II/Trigonometry but in increased depth and are challenged with more complex problems in class and on assessments.
     
    5 periods/cycle   
    Full-year course – 4 credits                                                                                                 
    Prerequisite(s): Geometry: Proof-Based or Geometry with Algebra Applications or the equivalent and permission of the department

  • Applied Precalculus

    This course prepares students for the concepts encountered in college precalculus. Students study polynomial, rational, exponential, logarithmic, and trigonometric functions in great detail.  Students are exposed to applications of key concepts and learn how to use various technological programs, such as the graphing calculator, to solve problems associated with these functions. This course covers much of the same material as in Precalculus, but some topics are explored in less depth. This course does not fully prepare students to take an advanced calculus in the following year. 

    5 periods/cycle   
    Full year course – 4 credits
    Prerequisite(s): Algebra II/Trigonometry or the equivalent and permission of the department

  • Precalculus

    The purpose of Precalculus is to prepare students for a rigorous approach to calculus in terms of dexterity with relevant algebraic and trigonometric skills and in terms of mathematical reasoning and problem-solving capabilities. Various functions are introduced, and an exploration of the properties of their graphs is included. Some topics studied in depth are analytic geometry, linear regressions, trigonometric and circular functions, exponential and logarithmic functions, polynomial and rational functions, continuity, and limits. 
    5 periods/cycle   
    Full-year course – 4 credits 
    Prerequisite(s):  Algebra II/Trigonometry or Algebra II/Trigonometry A and permission of the department          

  • Precalculus A

    Precalculus A is a rigorous course that prepares students for Advanced Calculus I & II. Students build on previous algebraic concepts to strengthen their conceptual understanding of mathematics. They explore proofs of mathematical concepts and examine multiple representations of functions and models. The course includes an in-depth study of functions, analytic trigonometry, polar and parametric equations, sequences, and series. Students utilize technology to examine patterns, formulate conjectures, and create mathematical models. 
    5 periods/cycle   
    Full-year course – 4 credits 
    Prerequisite(s): Algebra II/Trigonometry or Algebra II/Trigonometry A and permission of the department

  • Fundamentals of Calculus

    This year-long course will help students build a foundation from which they will be prepared to take an introductory college calculus course.  Course topics include limits, continuity, derivatives, differentiation, and integration; this narrow focus allows students to work at a supportive pace.  May include mathematical applications through the lens of business, physics, and more.  Students will be assessed via traditional assessments as well as self-paced and collaborative projects.  

    5 periods/cycle   
    Full year course – 4 credits
    Prerequisite(s): Precalculus A, Precalculus or Applied Precalculus and permission of the department

  • Advanced Calculus I

    This course will cover all of the main topics included in the first semester of college calculus. Students will be able to dive more deeply into the main concepts learned in a traditional calculus class. Subject matter covered in the first semester of the course will include functions and graphs, limits and continuity, the definition of a derivative, analyzing derivative graphs and differentiating basic functions. Subject matter covered in the second semester will include antiderivatives, evaluating basic integrals, applications of the definite integral limited to particle motion, net change, area and volume. The students will be introduced to differential equations and slope fields. There will be no required time outside of class for students to take tests.
     
    5 periods/cycle 
    Full year course – 4 credits
    Prerequisite(s): Precalculus A or Precalculus and permission of the department

  • Advanced Calculus I & II

    This fast-paced and challenging course covers many of the topics covered in the first and second semesters of college calculus. There are three major parts of this course: the derivative, including differentiation, applications of the derivative, and local linearity; the integral, including antiderivatives, differential equations, slope fields, Euler's Method, and applications of the definite integral; and Taylor Series, including polynomial approximations of transcendental functions and convergence tests. Concepts will be presented from graphical, numerical, and algebraic points of view.

    5 periods/cycle   
    Full-year course – 4 credits                                                                                                     
    Prerequisite(s): Precalculus A or Precalculus and the permission of the department


  • Advanced Calculus III

    The course begins by revisiting and extending topics from the Advanced Calculus I & II curriculum including numerical integration, techniques of antidifferentiation, differential equations, polar and parametric functions, and Taylor series.  Students learn about vector geometry and cylindrical and spherical coordinates, and the calculus of multivariable functions, including partial derivatives, the gradient, double and triple integrals, line and surface integrals, Green’s Theorem, and Stokes’ Theorem.  The syllabus covers the topics in a standard multivariable course.
     
    5 periods/cycle   
    Full-year course – 4 credits
    Prerequisite(s): Advanced Calculus I & II or the equivalent and permission of the department

  • Advanced Calculus II

    This spring semester course runs concurrently with the spring semester of Calculus I & II.  It covers applications of the definite integral, the calculus of polar and parametric defined functions, and series.  To enter this class, the student will have to do independent work (after completing Advanced Calculus I) on integration techniques, improper integrals, logistic growth and Euler’s Method.

    5 periods/cycle
    One semester course (Spring) – 2 credits 
    Prerequisite(s):  Advanced Calculus I or equivalent, successful completion of required independent study work, and permission of the department   

  • Statistics: Epidemiology

    This course will introduce students to the quantitative methods of scientific inquiry.  Epidemiology is the study of health conditions in a population - every citizen should know how to interpret its results.  Students will learn the history of the science of epidemiology and they will learn how to conduct and analyze observational studies.  Students will also learn how to design randomized clinical trials. Students will learn about diseases, transmission rates and ways that diseases are controlled.  Students will explore case studies and consider important ethical issues such as: genetic testing, animal experimentation and use of placebo treatments. Probability and its applications to understanding tests of statistical significance will be explored.
     
    5 periods/cycle
    One semester course (Fall) – 2 credits                                                             
    Prerequisite(s): Algebra II/Trigonometry or permission of the department

  • Statistics: Social Justice

    This course will develop the statistical skills needed to explore important social problems quantitatively.  Students will learn how to perform exploratory data analysis to detect patterns in data sets. They will learn how sampling distributions work in order to conduct and analyze surveys.  Students will gain an understanding of confidence intervals as they learn ways to reduce the margin of error. Students will research the impact of the Great Migration of Americans from the south to the north during the 20th Century by working with census microdata.  Linear regression models will be utilized to explore numerical relationships that describe social concerns such as racism, healthcare inequity, prison reform, environmental climate change and military spending.  Though we will extend some of the skills learned in Epidemiology, students who haven't taken the first semester Statistics course will be brought up to speed in the first two weeks.
     
    5 periods/cycle
    One semester course (Spring) – 2 credits                                                             
    Prerequisite(s): Algebra II/Trigonometry or permission of the department

  • Advanced Statistics

    This challenging course in statistics follows the curriculum covered in a college level statistics and data science course. The course includes an solid introduction to R programming, data visualization, descriptive statistics and the basics of statistical modeling (e.g. one-way ANOVA and simple regression). The approach to inferential statistics is grounded in simulation, randomization and bootstrapping. Students perform frequent projects that include bivariate and multivariate models. Students will learn to apply statistical methods appropriately while refining their communications skills. 


    5 periods/cycle   

    Full year course – 4  credits                                                                                                               

    Prerequisite(s): Permission of the department 

  • Financial Mathematics

    This semester-long course will help students enhance their financial awareness and feel empowered to make financial decisions.  Topics include spending and saving, credit and debt, employment and income, investing, and risk management and insurance. Mathematical applications include simple and compound interest calculations, budget creation, and fee analysis.  This course is aligned with the National Standards in Personal Finance Education.  
    This is a possible course for 23-24 and its scheduling will depend on student interest and classroom and teacher availability. Students who would like to request this course should also select an 'Alternate Request' in case this course does not run.
    5 periods/cycle
    One-semester course (Fall) – 2 credits                                                             
    Prerequisite(s): Algebra II/Trigonometry and permission of the department


  • Principles of Geographic Information Systems

    This course will explore two-dimensional representations of our three-dimensional world. Students will create hand-drafted maps, work with physical maps, and utilize digital maps to create their own maps of New York City. Students will gain an understanding of geographic information systems (GIS) and learn how to create geographic databases used to analyze important issues such as climate change, social inequities, or the future of our city.
    Students will build on the mathematical foundations of geometry and trigonometry to learn basic concepts of spherical trigonometry, Non-Euclidean geometries, and graph theory. A review of similarity and an introduction to projections from the spheroid Earth to two-dimensional maps are integral components of the course. 
    This class could serve as a senior project class.
    5 periods/cycle
    One-semester course  – 2 credits                                                             
    Prerequisite(s): Algebra II/Trigonometry and permission of the department
Friends Seminary actively promotes diversity, equity, inclusion, and anti-racism in all its programs and operations, including admissions, financial aid, hiring, and all facets of the educational experience. To form a community which strives to reflect the world’s diversity, we do not discriminate on the basis of race or color, religion, nationality, ethnicity, economic background, physical ability, sex, gender identity or expression, or sexual orientation. Friends Seminary is an equal opportunity employer.
FRIENDS SEMINARY
222 East 16th Street
New York, NY 10003
P: 212-979-5030
F: 212.979.5034
Friends Seminary — the oldest continuously operated, coeducational school in NYC — serves college-bound day students in Kindergarten-Grade 12.