Math
Mathematics courses at George School serve a variety of purposes. They stimulate the imagination while developing and polishing basic skills. They prepare many students for the study of higher-level mathematics in college while acquainting all students with a broad array of mathematical methods and computing technology. Our goal is to help students develop the ability to think mathematically—to analyze, to synthesize, and to hypothesize—making use of mathematics both as a language and as a methodology.
The department offers courses ranging from basic algebra to beyond calculus. In order to provide appropriately challenging and supportive classes, students are grouped according to mathematical facility and interest. Course recommendations and decisions about placement are based primarily on a student's previous coursework, recent performance, and future plans.
Algebra 1
The fundamental mathematical practice of using variables to explore and describe patterns is introduced in this course. Topics covered include the evaluation and manipulation of algebraic expressions, the solution and graphical representation of linear equations and inequalities, the solution of systems of equations in two variables, the solution of quadratic equations by factoring and through use of the quadratic formula, and the algebraic solution of word problems. A new concept is presented almost daily. Review is built in as needed. Daily homework includes problems similar to those worked in class as well as problems designed to stretch students' understanding. Facility with arithmetic operations is assumed. While this course is designed for students who did not take Algebra 1 in junior high, it is also appropriate for students who have had an Algebra 1 course but would benefit from further grounding in the subject prior to enrolling in a second-year algebra course.
Intermediate Algebra
This course begins with a rapid review of Algebra 1 skills and builds on them to cover all of the topics of Algebra 2. Sequences and series, operations with matrices, and introductory descriptive statistics are covered as well. Students must develop effective note-taking skills since there is no textbook. Typically a new topic is introduced each day. Daily homework includes problems similar to those worked in class in addition to problems designed to stretch students' understanding. Students are expected to identify areas in which they might need more help or practice to master a skill that is presented in class or in homework.
Prerequisite: An Algebra 1 course including graphing linear equations and solving quadratic equations
Advanced Algebra
This course covers all of the topics of Algebra 2 without reviewing Algebra 1. It also includes the study of vectors, parametric equations, sequences and series, elementary statistics, and topics in number theory. By modeling a creative mathematical process themselves, students come to appreciate mathematics as a collection of ideas that people invent. The class does not use a textbook and homework typically includes thought-provoking problems with methods of solution that have not been demonstrated by the teacher. During class, the teacher guides discussion and provides a framework which enables students to learn new techniques. Students write their own "books" by taking notes during discussions. Because many students who take this course have not previously had to study to do well in math, attention is given to techniques for efficient and effective learning. This course is taught at a rapid pace. Students are encouraged to develop the confidence to risk failure by tackling questions that stretch their understanding in class, on homework, and on tests.
Prerequisite: Strong performance in an Algebra 1 course including graphing linear equations and solving quadratic equations
Geometry
Two- and three-dimensional figures are studied in this course, with an emphasis on concrete, numerically based examples and frequent hands-on activities. Topics covered include area, volume, congruence, similarity, compass and straightedge constructions, the Pythagorean Theorem, and trigonometric ratios. The ability to generalize and characterize a pattern algebraically from specific cases is developed. Students explore inductive and deductive reasoning patterns and begin to develop the ability to present mathematical arguments. Proof writing is not a major emphasis of the course. A new concept is presented almost daily. While many daily homework problems are similar to problems that have been worked through in class, others require students to apply what they know to new types of problems; the pace enables detailed discussion of all assigned homework problems. Algebraic topics are reviewed on an as-needed basis.
Prerequisite: Algebra 1 or Intermediate Algebra
Geometry with Proofs
All of the topics of Geometry are covered at a faster pace and in greater depth. The course material is extended to include proof writing, coordinate geometry, transformations, and the Laws of Sines and Cosines, all of which help to form a strong foundation for precalculus. As time allows, additional topics might include the Golden Ratio, taxicab geometry, graph theory, and fractals. A new lesson is encountered daily through a lecture, group investigation, or an independent project. While many daily homework problems are similar to problems that have been worked through in class, others require students to apply what they know to new types of problems. Strong graphing and note-taking skills are assumed.
Prerequisite: Advanced Algebra, Intermediate Algebra, or Algebra 1 (A- or better)
Abstract Geometry
This course covers all the topics of Geometry with Proofs and much more. It is designed to encourage students to contemplate and appreciate the nature of mathematics, both as an axiomatic deductive system and as a science of patterns. The concept of proof is central to the course. The emphasis on creative problem solving, clear thinking, and careful articulation provides an important foundation for advanced mathematics courses at George School and beyond. Advanced topics include concurrency proofs, Fermat primes, theorems from Ceva and Heron, and circular functions. To lay the foundation for IB HL mathematics, trigonometry is introduced, vectors are reviewed and extended to three dimensions, and the idea of a limit is informally introduced, applied, and used to derive formulas. Daily homework requires students to apply concepts discussed in class to new problems in a creative fashion. Strong graphing, note-taking, and algebraic skills are assumed, as is the ability to generalize a pattern from specific cases. A year of geometry at the junior high level is helpful, but not required.
Prerequisite: Intermediate Algebra (A- or better) or Advanced Algebra (C or better)
Algebra 2
A thorough review of Algebra 1 skills is intertwined with the development of more advanced algebraic skills in this course. Students are introduced to the concept of a mathematical function and they do extensive work with linear and quadratic functions and their graphs. Quadratic equations with complex roots are considered and quadratic inequalities are explored. Logarithmic and exponential expressions, equations, and functions are introduced. Students deepen their understanding of rational, absolute value, and polynomial expressions and equations. Concepts are introduced or extended almost daily. Review is built in as needed. Daily homework problems are similar to problems worked through in class.
Prerequisites: Algebra 1 (C-), Geometry, or Geometry with Proofs
Algebra 2 with Trigonometry
Extending the skills developed in Algebra 1 and Geometry with Proofs, this course introduces new algebraic concepts that include rational expressions and equations; quadratic expressions, equations and inequalities; complex numbers; sequences and series; the binomial theorem; and basic matrix operations.There is a thorough discussion of the concepts relating to functions, including transformation of graphs and inverse functions. Polynomial, absolute value, logarithmic, and exponential expressions and functions are also studied. The course concludes with a review of right triangle trigonometry and an introduction to radian measure, the unit circle, and graphs of circular functions. New topics are introduced daily. While many daily homework problems are similar to problems worked in class, others require students to apply what they know to new types of problems. Strong graphing, factoring, and note-taking skills are assumed. Students who completed Intermediate Algebra with a grade of B- or lower should take this course prior to a precalculus course.
Prerequisites: Algebra 1 (B+), Geometry with Proofs (C), or Geometry (A-)
Functions Trigonometry and Statistics
This exploration-based class focuses on a different mathematical theme each term and includes real-world applications of the skills developed. During the first term, students review and extend the study of functions and relations begun in Algebra 2, with particular attention to translations and transformations of polynomial and exponential functions. The second term is devoted to trigonometry, including radian measure, the unit circle, the graphs of the six circular functions, and translations and transformations of these graphs. The third term provides an introduction to probability and statistics. The class explores permutations and combinations, games of chance, independent events, and conditional probability. Techniques of descriptive statistics are discussed, including stem and leaf plots, box and whisker diagrams, frequency histograms, linear regression, correlation, and the normal curve. The pace is relaxed, yet purposeful. If a specific exploration is proving especially fruitful mathematically for a particular class, it might be extended even if that means not covering every topic on the original syllabus.
Prerequisite: Algebra 2 (C-) or Algebra 2 with Trigonometry
IB Math Studies SL
All students in this course are expected to take the IB Math Studies exam.
This course covers a variety of mathematical topics, including number and algebra; sets, logic, and probability; linear, quadratic, exponential, sine and cosine functions and their graphs; right triangle trigonometry; descriptive and introductory inferential statistics; financial math and introductory differential calculus. Students complete a major project which also serves as the internal assessment portion of the IB exam. The capacity for independent work is important to a student's success.
Prerequisite: Either of the following options
1. Geometry with Proofs (C+) together with one of Intermediate Algebra (C+), Algebra 2 with Trig (C+), or Algebra 2 (B+).
2. Functions, Trigonometry, and Statistics (B+)
IB Math SL 1—Precalculus
This course is the first in a two-year sequence that prepares students for the calculus-based IB Mathematics SL exam. The concept of a function is the central theme of this course. Concepts covered include domain and range, composition, translation, transformation, and inverse functions. A primary goal is to help students learn to shift fluently between algebraic and graphical representations of functions. Polynomial, exponential, logarithmic, and trigonometric functions are studied in depth and the concept of a limit is introduced. Additional topics include sequences and series, vectors, and matrices.
A strong working knowledge of linear and quadratic functions is assumed. In addition, students are expected to have good algebraic skills, good graphing skills, and familiarity with right triangle trigonometry. While many daily homework problems are similar to problems worked in class, others require students to apply what they know to new types of problems. The capacity for independent work is important to a student's success. Students begin to develop an IB mathematics portfolio during this class. The portfolio is completed in the second course of the sequence, IB Math SL 2: Calculus.
Prerequisite: Any of the following five options
1. Either Intermediate Algebra (B or better) or Advanced Algebra together with either Geometry with Proofs (B or better) or Abstract Geometry.
2. Algebra 2 with Trigonometry (B– or better)
3. Algebra 2 (A)
4. Functions, Trigonometry, and Statistics (A)
5. IB Math Studies
IB Math HL 1—Precalculus
Students in this course spend the first two terms studying traditional precalculus topics, including trigonometry from a functional point of view, theories of polynomial equations, logarithmic and exponential functions, inverse functions, complex numbers including applications of DeMoivre's theorem, polar coordinates, vectors in three dimensions, probability, and basic linear algebra. The third term is devoted to a study of the statistical topics from the IB Math HL syllabus. Students begin to develop an IB mathematics portfolio during this class. The portfolio is completed in IB Math HL 2: Calculus. The pace is very fast. Because the class frequently takes the form of a Socratic dialogue with questions asked and solutions offered by both teacher and students, it is imperative that students develop the courage to write down and share their ideas.
Prerequisites: One of the following two options:
1. Either Advanced Algebra (C) or Intermediate Algebra (A-) together with either Abstract Geometry (C) or Geometry with Proofs (A-)
2. Algebra 2 with Trigonometry (A)
IB Math SL 2—Calculus
The fundamentals of differential and integral calculus are covered in this course. Topics include limits; continuity; understanding derivatives as functions, slopes, and rates of change; derivatives of polynomial, rational, trigonometric, exponential, and logarithmic functions; analysis of graphs; optimization; related rates; rectilinear motion; anti-differentiation; the Fundamental Theorem of Calculus; integration by substitution; and applications of integration to area, volume, rectilinear motion, and accumulation problems. Topics in statistics introduced in SL1 are reviewed and extended. These include discrete random variables and normal distributions. Students complete an IB mathematics portfolio in this class. Each day in class the homework is reviewed and questions are answered. New concepts are presented with examples, in preparation for the next night's homework. Student input and questions drive class discussion. Strong algebraic and graphing skills are assumed. While students are not required to take the IB exam, they are welcome to do so.
Prerequisite: An SL precalculus course (B) or IB Math HL 1
AP Calculus AB
This course covers all topics included in the College Board syllabus. Throughout the course, problems are considered from graphical, numerical, and analytical perspectives with an aim toward developing students' ability to shift easily from one perspective to another. There is an emphasis on learning to understand, use, and appreciate the value of the precise technical language (definitions, theorems, etc.) of mathematics. An awareness of the historical context of the development of calculus and an appreciation of its importance as a human achievement are cultivated. Students learn to discern situations in which technology can be a helpful tool in the solution of a problem. Graphing calculators are used extensively. The pace is fast. Students are expected to work as mathematicians do in that they are asked frequently to try problems without having been explicitly taught how to find the solutions. Excellent algebraic, graphing, and organizational skills are assumed, as is a very good understanding of trigonometric functions. Students are required to take the AP exam. (IB diploma candidates should take one of the calculus courses with IB in the title rather than this one unless they have taken Precalculus for IB Math Studies.)
Prerequisites: IB Math HL 1 (C) or an SL precalculus course (A)
IB Math HL 2—Calculus
This course covers all calculus topics included in the IB Mathematics HL core syllabus plus the topics from the HL Calculus option. Throughout the course, problems are considered from graphical, numerical, and analytical perspectives with an aim toward developing students' ability to shift easily from one perspective to another. There is an emphasis on learning to understand, use, and appreciate the value of the precise technical language (definitions, theorems, etc.) of mathematics. Students learn to discern situations in which technology can be a helpful tool in the solution of a problem. Graphing calculators are used extensively. Students are required to complete an IB portfolio. The pace is intense. Students are expected to work as mathematicians do in that they are asked frequently to try problems without having been explicitly taught how to find the solutions. Excellent algebraic, graphing, and organizational skills are assumed, as is a very good understanding of trigonometric functions and a working knowledge of the statistics covered in IB Math HL 1. All students are required to take either the IB Math HL exam or the IB Math SL exam. (Students are also able to take the AP Calculus (AB) exam if they so choose as the course covers substantially more calculus than the AP Calculus (AB) course.)
Students are required to complete a summer assignment in preparation for class.
Prerequisite: IB Math HL 1 (A)
Statistics
By the end of this course, students should be able to understand and to appropriately use the terminology and symbols of statistics; formulate questions that can be addressed with data; collect, organize, and display relevant data to answer statistical questions; select, use, and evaluate descriptive methods to analyze data; understand and apply basic concepts of probability; and critique graphs and descriptive data analyses presented in newspapers and magazines. Concepts include graphical methods, descriptive analyses of univariate and bivariate data, probability, and probability distributions. Some concepts of inferential statistics are also included. Students learn how to perform analyses using paper and pencil, a statistical calculator, and the computer, with an emphasis on the interpretation of results. The pace is relaxed, yet purposeful. An independent project will be assigned; if IB students are in the class, the project will be assigned to meet the requirements for the IB math studies project.
Prerequisite: Algebra 2 (B-) or Algebra 2 with Trigonometry (C-) or Functions, Trigonometry, and Statistics (C-)
AP Statistics
This course follows the College Board syllabus, which includes all of the topics covered in Statistics plus concepts of variation, especially as related to statistical inference, sampling distributions, estimation and confidence intervals, and hypothesis testing at least through two-sample t-tests. Students learn how to perform analyses using paper and pencil, a statistical calculator, and the computer, with an emphasis on the interpretation of results. Class activities consist of lecture, problem solving, and group discussion, with a heavy emphasis on analytical discussion. The pace is rapid and the topics are complex. Students are expected to be inquisitive about data, analyses, and interpretation and to contribute their thoughts actively to class discussions. Readings and homework are assigned daily. Students are expected to spend at least an hour on homework for each class meeting; many students find that it takes more than an hour to do a thorough job. Students are expected to take the AP exam. Students complete an independent research project at the end of the year. Students are required to complete a summer assignment in preparation for class.
Prerequisite: An IB SL math course (B) or IB Math HL 1 (C-)
Computer Programming and Robotics
This course is cross-listed in the math and science departments. Students use the BasicX language to design autonomous robotics applications for wheeled, walking, and facially-expressive robots manufactured by Robodyssey Systems. It is assumed that students are already comfortable with computer technology but know very little about computer programming. Topics covered include top-down and event-driven programming, logical statements, loops, arrays, sensor input, motor control, relays, and GPS programming. All students have the opportunity to enter a robot in local or national competitions such as firefighting, soccer, and dance. Near the end of the year, desktop programming is introduced via video game programming using the Visual Basic.NET language. This is an applied science and mathematics course for students with various academic backgrounds. The course gives students who have mastered trigonometry and algebra an opportunity to use their knowledge to create complex computer algorithms. Students who are familiar with, but have not mastered these mathematical skills, can use these ideas in practical and relevant ways to help refine and augment the science and mathematics curricula. This is a project-oriented course and is largely driven by student interests.
This course fulfills the physical science requirement.
Prerequisite: A precalculus course (can be taken concurrently) or Advanced Algebra (B) or Algebra 2 with Trigonometry (B)