Mathematics

William Darling, Department Chair

Math has a central place in our daily lives (how do you double a recipe or figure out sales tax?) and in many of our most significant innovations. Suspension bridges, airplanes, robotic surgery, and the computer algorithms that bring us social media all rely on creative applications of math. As we ready students for a future we cannot entirely predict, we are confident that quantitative reasoning, creative problem solving, and simple hard work will remain essential.

Friends’ Central’s math department combines a belief in the ability of our students to rise to intellectual challenges and a program that offers a high degree of flexibility in course selection and sequence. For example, students can design a sequence that includes a year of calculus without doing additional summer work. More advanced students have the opportunity to progress through the third level of calculus within our curriculum. Friends’ Central facilitates independent study or enrollment in math classes at nearby colleges and universities for students who exceed the third level of calculus.

All students are required to take three math courses; the majority take one each year. Most classes have both a regular and an advanced level offered. Some students choose to accelerate by taking a summer course. A typical Upper School path includes algebra II, precalculus, and calculus.

While many students take two years of calculus before graduating, others choose statistics over a second year of calculus. We also offer a third year of calculus (Topics in Multivariable Calculus and Linear Algebra).

We have an active math team, and many students compete successfully in national and international mathematics competitions.

Algebra I

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.)

Geometry

The geometry courses reflect the standards proposed in the Common Core curriculum. Topics receiving attention include polygons, triangle congruence and similarity, perimeter and area, surface area and volume, and geometric constructions. Instead of a traditional emphasis on two-column proof, this course focuses on a more general understanding of logical reasoning. Prerequisite: Algebra I

Geometry Advanced

This course covers all topics listed in Geometry but does so in greater depth. In addition, there is a much greater
emphasis on two-column and paragraph proofs. Prerequisite: Algebra I and recommendation of the department

Algebra II

Topics in algebra are explored with a greater degree of sophistication than in Algebra I. New topics include an introduction to sequences and series, extensive work with quadratic functions including complex numbers, place value and the binary number system, exponential functions and relations, graph transformations, exponents, and data analysis. The TI Nspire graphing calculator app is used throughout the course. Prerequisite: Algebra I

Algebra II Advanced

Advanced Algebra II will explore the topics discussed in Algebra II at a deeper level and without review of Algebra I topics. Additionally, Advanced Algebra II will include a variety of topics such as further study of complex numbers, logarithms, and matrix computation. The TI Nspire graphing calculator app is used extensively throughout the course. Prerequisite: Algebra I and recommendation of the department

Precalculus

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 and their applications through extensive use of the TI Nspire graphing calculator app. Students will also develop mathematical models for various “real world”
situations. Prerequisite: Geometry and Algebra II

Precalculus Advanced

The main focus of this course will be 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. The course will also introduce the study of probability, combinations, and, time permitting, conic sections and polar coordinates. Prerequisite: Geometry and Algebra II Advanced and recommendation of the department. There is a placement test for this course.

Calculus I

This course is intended to be an introduction to basic ideas of differential calculus and 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

Calculus I Advanced

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. 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 cover the basic derivative rules and integration rules for the functions that were studied in Precalculus and how to apply them. Prerequisite: Precalculus or Precalculus Advanced

Calculus II Advanced

This course is for students who have successfully completed a year of Calculus. The course will cover more advanced topics in single variable Calculus, such as advanced techniques of integration, parametric and polar coordinate graphing, infinite series, first-order ordinary differential equations and applications, and three-dimensional coordinate systems. Prerequisite: Calculus I Advanced or Calculus I

Statistics

This course is similar in description to Statistics Advanced. The major difference between Statistics and Statistics Advanced is that the units on exploring data, planning a statistical study, and anticipating patterns using probability will be given additional focus and attention, and the unit on confirming models through statistical inference will be deemphasized. Students in this course will have a greater opportunity to do “real world” projects in data collection and analysis than students in the advanced section. Prerequisite: Algebra II

Statistics Advanced

This course is equivalent to a one-semester, introductory, non-calculus based college course in statistics. The purpose of this course is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad themes: exploring data, planning a statistical study, anticipating patterns using probability, and confirming models through statistical inference. The course makes significant use of the TI NSpire app for statistics on the iPad 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

Calculus III: Topics in Multivariable Calculus & Linear Algebra

Part of the year is spent looking at topics in multivariable calculus: equations of lines and planes in 2 and 3 dimensions, partial derivatives, topics in vector calculus (dot products, cross products, projections), volumes and areas using multiple integrals, directional derivatives, and more. In addition to skill-building, the approach to learning includes time dedicated to proof-writing in class under the direction of the teacher. Conversations are informed and enriched by the breadth of the instructor’s mathematics base and are shaped by the students’ interests. Students also study introductory topics in Linear Algebra. Topics covered include dot/inner products, modular arithmetic, theory,and solution of systems of linear equations, spanning sets, linear dependence/independence, orthogonality, matrix operations, determinants, invertibility, subspaces, basis, dimension, and rank, plus applications and, if time and interest allow, an introduction to eigenvalues and eigenvectors. Investigation is followed by working of problem sets from the text, and proof remains a form of exercise. Prerequisite: Calculus II Advanced

Making Decisions: An Introduction to Economics

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. Themain 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: Algebra II