# Math Courses

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 describe patterns, explore relationships, and solve problems is the focus of this course. Particular attention is paid to developing an awareness of the difference between expressions and equations, an awareness of the relationships among mathematical operations, and an understanding of the connections between algebraic and graphical representations of pattern. Topics through which these central concepts are explored include linear equations and inequalities, compound inequalities in a single variable, systems of linear equations, quadratic expressions and equations, rational expressions and equations, radical expressions and equations, polynomial expressions, and expressions involving integer exponents.

This course is designed for those students who have not yet taken an algebra course as well as for those who have taken the first half of Algebra 1, but would benefit from further grounding in the material covered.

Open to: Freshmen

### Accelerated Algebra 1

The mathematical focus of this course is similar to that of Algebra 1. All of the topics of Algebra 1 are covered at a faster pace and in greater depth. Additional topics include compound inequalities in two variables, expressions involving rational exponents, linear regression, absolute value equations and inequalities, imaginary numbers, functions, and transformations of linear, quadratic, and absolute value functions.

Good arithmetic skills, a working knowledge of the Cartesian plane, and the ability to solve linear equations are assumed. This course is designed for students who have a good understanding of the material from a thorough pre-algebra course or the first half of an Algebra 1 course as well as for those who have had a full Algebra 1 course, but would benefit from a deepening of their understanding prior to enrolling in an Algebra 2 course.

Open to: Freshmen

### Geometry

Two- and three-dimensional figures are studied in this course, with an emphasis on concrete, numerically based examples. Topics covered include area, volume, congruence, similarity, 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.

Open to: Sophomores

Prerequisite: Algebra 1 or Accelerated Algebra 1

### Intensive Geometry with Trigonometry

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, compass and straightedge constructions, coordinate geometry, and transformations. To form a strong foundation for precalculus, the third term includes studies of right triangle trigonometry and the Laws of Sines and Cosines. 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.

Open to: Freshmen and sophomores

Prerequisite: Accelerated Algebra 1 (B+) or Algebra 1 (A)

### Advanced Algebraic & Geometric Analysis

This course is designed to challenge those 9th and new 10th grade students who have unusually strong backgrounds in algebra, geometry, or both. Students build on prior experience in algebra and geometry as a basis for investigating advanced topics. Emphasis is placed on individual and group exploration of mathematical ideas in order to solve unfamiliar problems, to discover patterns, and to prove results. Creative problem solving, clear thinking, and careful articulation provide an important foundation for advanced mathematics courses at George School and beyond. The concept of proof is central to the course; a wide variety of proof strategies are explored. Topics from earlier algebra courses that are studied in greater depth include functions (polynomial, absolute value, rational, radical, exponential, and logarithmic), inverse functions, complex numbers, and nonlinear inequalities. Topics from earlier geometry courses that are studied in greater depth include congruency axioms, similarity, parallel properties, area, perimeter, and volume. Topics which may be completely new to students include set theory, vectors in two and three dimensions, parametric equations, the binomial theorem, number theory, algebraic proof, matrices, descriptive statistics, conic sections, sequences and series, Fermat primes, and theorems from Ceva, Heron, and Apollonius.

The class uses a problem book rather than a textbook. Daily homework requires students to creatively apply concepts discussed in class to thought-provoking problems with methods of solution that may not have been demonstrated by the teacher. 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 of advanced mathematics. This course is taught at a rapid pace. Students are encouraged to develop the confidence to risk failure by tackling questions that deepen their understanding in class, on homework, and on tests. Strong graphing and algebraic skills are assumed, as is the ability to generalize a pattern from specific cases.

Open to: Freshmen

Prerequisite: Strong performance in an Honors Algebra 1 course including solving and graphing quadratic equations, and at least some exposure to concepts typically taught in high school geometry and Algebra 2. While a full course in geometry or Algebra 2 is helpful, it is not required.

### 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.

Open to: Sophomores and juniors

Prerequisites: A geometry course together with either Algebra 1 (C-) or Accelerated Algebra 1

### Intensive Algebra 2

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; and the binomial theorem. 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. Additional topics might include probability and combinatorics, statistics, and basic matrix operations. 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.

Open to: Sophomores and juniors

Prerequisites: One of the following two options:
1. Intensive Geometry with Trig (C)
2. Geometry (A-) together with Algebra 1 (A) or Accelerated Algebra 1 (B)

### Functions, Trigonometry & 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, exponential, and logarithmic 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 fruitful for a particular class, it might be extended even if that means not covering every topic on the original syllabus.

Open to: Juniors and seniors

Prerequisite: Algebra 2 (C-) or Intensive Algebra 2

### IB SL Mathematics: Applications & Interpretations

This class was formerly called SL Math Studies. The topics covered in this survey course are those of the IB Math Studies syllabus, including descriptive and introductory inferential statistics; geometry and trigonometry; unit conversion; mathematical models (linear, quadratic, exponential, and rational); introductory differential calculus; sets, probability, and logic. The approach taken emphasizes the development of mathematical reasoning skills and the understanding of fundamental concepts. Most topics are explored in a real-world context. Students design and complete an independent data-based research project which also serves as the internal assessment portion of their IB grade. Solid algebraic skills and the capacity for independent work are important to a student's success.

Open to: Juniors and seniors

Prerequisite: Either of the following options: 1. Intensive Geometry with Trig (C+) or Geometry (A-), together with either Algebra 2 (A-) or Intensive Algebra 2 (C+) 2. Functions, Trigonometry, and Statistics (B+)

### Intensive Precalculus

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.

Open to: Sophomores, juniors, and seniors

Prerequisite: Any of the following four options:
1. Advanced Algebraic and Geometric Analysis
2. Algebra 2 (A)
3. Intensive Algebra 2 (B)
4. Functions, Trigonometry, and Statistics (A)
5. IB Math Studies (B)

### Advanced Precalculus with Discrete Math

Students in this course develop their ability to investigate a problem mathematically and hone their proof-writing skills by exploring such topics as trigonometric functions, theories of polynomial equations, logarithmic and exponential functions, inverse functions, complex numbers, DeMoivre's theorem, polar coordinates, vectors in three dimensions, probability, combinatorics, linear algebra, and mathematical induction. 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 have or develop the courage to write down and share their ideas. Students who take this course in their sophomore year may plan on taking Advanced Placement (AP) Calculus as juniors and a higher level International Baccalaureate (IB) mathematics course as seniors. Students who take the course in their junior year may plan on taking AP Calculus or a standard level IB mathematics course as seniors.

Open to: Sophomores, juniors, and seniors

Prerequisites: Either Advanced Algebraic and Geometric Analysis (C) or the combination of Intensive Geometry with Trigonometry (A) and Intensive Algebra 2 (A)

### AP Computer Science A

This course is cross-listed in the math and science departments. AP Computer Science A is an introductory course in computer science for those who already have some basic programming experience. The course emphasizes object-oriented programming methodology with a concentration on problem solving and algorithm development, and is the equivalent of a first semester college-level course in computer science.
The central activity of the course is the design and implementation of computer programs to solve problems; the goal of the course is to develop and hone skills that are fundamental to the study of computer science. Creating computer programs is used as a context for introducing other important aspects of computer science, including the development and analysis of algorithms, the development and use of classes and fundamental data structures, the study of standard algorithms and typical applications, and the use of logic and formal methods. The responsible use of these systems is an integral part of the course.
The computer language studied is Java, as required by the AP curriculum. The prerequisites for entering this course include knowledge of algebra, a foundation of mathematical reasoning, and experience in problem solving. In addition, because documentation plays a central role in the programming methodology, competence in written communication is a requirement. It is expected that all students in the course will sit for the AP Computer Science A exam, which is administered in the spring semester.

Open to: Juniors and seniors

Prerequisite: Intensive Physical Computing and Robotics (A) or placement test

### IB SL Calculus

The topics covered in this survey course are those of the IB SL Mathematics: Analysis and Approaches syllabus. 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.

Open to: Juniors and seniors

Prerequisite: One of the following options: 1. Intensive Precalculus (B) 2. Advanced Precalculus with Discrete Math

### AP Calculus–AB

This course covers all topics included in the College Board syllabus for AP Calculus AB. It is designed to be the equivalent of a college-level Calculus 1 course. 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 taking this course in 11th grade must follow it with IB HL Mathematics, while IB diploma candidates taking this course in 12th grade must have previously taken IB SL Calculus.)

Open to: Juniors and seniors

Prerequisites: Advanced Precalculus with Discrete Math (C), or Intensive Precalculus (A)

### AP Calculus–BC

This course covers all topics included in the College Board syllabus for AP Calculus BC. It is designed to be the equivalent of college-level Calculus 1 and 2 courses. Because of this, the course moves extremely quickly, and the Calculus 1 material is covered at a particularly fast pace. 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. 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 taking this course in 11th grade must follow it with IB HL Mathematics and/or IB HL Further Mathematics, while IB diploma candidates taking this course in 12th grade must have previously taken IB SL Calculus.)

Open to: Juniors and seniors

Prerequisites: One of the following options:
1. Advanced Precalculus with Discrete Math (A-) or permission of the department
2. IB SL Calculus (A-)
3. AP Calculus AB (B-)

### IB HL Mathematics

The topics covered in this survey course are those of the IB HL Mathematics: Analysis and Approaches syllabus, which includes all of the content of the SL Analysis and Approaches curriculum. Unlike AP math exams which focus exclusively on either calculus or statistics, the IB HL Mathematics exam requires an advanced level of mastery of a wide range of mathematical topics. Students in this course complete their study of the HL Mathematics curriculum by covering those syllabus topics which are not part of Advanced Algebraic and Geometric Analysis, Advanced Precalculus with Discrete Math, or AP Calculus - AB. The primary areas of focus are advanced integration techniques, differential equations, the calculus of sequences and series, and the probability and statistics portion of the HL Mathematics syllabus. 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 exploration. 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 trigonometry. All students are required to take the IB HL Math exam.

Students are required to complete a summer assignment in preparation for class.

(Not offered in 2020-21)

Open to: Seniors

Prerequisite: Advanced Precalculus with Discrete Math (B) and AB Calculus (B) or BC Calculus (B)

### 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, probability distributions, and sample and experimental design. Topics in term three may include inferential statistics and financial literacy. Students learn how to perform analyses using paper and pencil, and a statistical calculator, with an emphasis on the interpretation of results. A written independent project may be assigned.

Open to: Juniors and seniors

Prerequisite: One of the following five options:
1. Algebra 2 (B-)
2. Intensive Algebra 2 (C-)
3. Functions, Trigonometry, and Statistics (C+)
4. IB Math Studies
5. A precalculus course

### 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.

Open to: Juniors and seniors

Prerequisites:
1. IB SL Math Studies (B) or Intensive Precalculus (B) or Advanced Precalculus with Discrete Math (C-)
2. Two grades of B or higher in life science or mainstream history courses

### Further Mathematics: Discrete Mathematics & Number Theory

This course provides a proof-based introduction to elementary number theory and discrete mathematics. Topics from number theory include the Euclidean algorithm and prime factorization, congruences and modular arithmetic, Fermat’s and Wilson’s theorems, as well as quadratic residues, Gaussian integers, and cryptography. Discrete topics include sets and induction, recurrence relations, probability, graph theory, and combinatorial geometry. Depending on student experience, the course may integrate the knowledge of calculus to provide a limit-based analytic approach to number theory. This is a fast-paced class. This course appeals to those students intending to major in a field involving significant post-calculus mathematics, though any student with an interest in mathematics for its own sake will be well-served. This course will be offered in alternate years, beginning in the 2020-2021 school year. Students will need to have demonstrated significant mathematical maturity and a willingness to confront abstraction.

Students are required to complete a summer assignment in preparation for this class, which may vary depending on their background. In particular, senior IB Diploma Candidates taking this class in lieu of HL Mathematics Analysis and Approaches will need to complete an assignment particular to that content.

Open to Seniors.

This course is offered 2020-2021 and alternate years.

Prerequisites: One of the following two options: 1. Advanced Precalculus with Discrete Mathematics (A), concurrent registration in BC Calculus, and departmental approval, or 2. BC Calculus (B) and departmental approval. Open to: Seniors

### Further Mathematics: Linear Algebra

This course is an introduction to the tools and techniques of linear algebra. Topics studied include vectors and matrices, vector spaces and dimension, determinants, eigenvalues, matrix factorization, and applications to graphs, dynamics, and probability. Where appropriate, students will use scientific computation tools to enrich their understanding of the concepts studied and to handle massive datasets. Proofs are an essential component of the class. This course appeals to those students intending to major in a field involving significant post-calculus mathematics, though any student with an interest in mathematics for its own sake will be well-served. This course will be offered in alternate years, beginning in the 2021-2022 school year. Students will need to have demonstrated significant mathematical maturity and a willingness to confront abstraction.

Students are required to complete a summer assignment in preparation for this class, which may vary depending on their background. In particular, senior IB Diploma Candidates taking this class in lieu of HL Mathematics Analysis and Approaches will need to complete an assignment particular to that content.

Open to Seniors.

This course is offered 2021-2022 and alternate years.

Prerequisites: One of the following two options: 1. Advanced Precalculus with Discrete Mathematics (A), concurrent registration in BC Calculus, and departmental approval, or 2. BC Calculus (B) and departmental approval. Open to: Seniors

### Physical Computing & Robotics

This course is cross-listed in the math and science departments. This is a self-directed course that is project-oriented and driven largely by student interests. Students build their own PRT3 motherboard and learn to use the Arduino Language to program their own Teensy 3.2 microcontroller development board. (The Arduino language is based heavily on the well-known C‑programming language.) It is assumed that students are already comfortable with computer technology but know very little about computer programming. Throughout the year, students create autonomous robotics applications for wheeled, walking and facially-expressive robots manufactured by Patton Robotics, or they can design and build their own robot or embedded controller system.

Programming topics include logical statements, functions, loops, recursion, sensor input, servomotor programming, and actuator control. While students in this class are permitted to use some algorithms found online or authored by other students, they are required to write some algorithms from scratch. Students will be taught how to use computer-aided design (CAD) software to make 2D and 3D models which can then be cut with the 100-watt laser cutter or printed on the many 3D printers in the laboratory. Students are also introduced to electronics, circuit design, soldering, and mechanical engineering concepts and tools. Once the student has shown an understanding of programming basics, CAD design, and electronics, they are on their own to create, build and program one or more robotic applications that will perform some autonomous task, usually incorporating simple feedback control systems. Students will display their projects during the Interactive Robotics Open House, which takes place at the end of the academic year.

Students who are uncomfortable applying science and mathematics to everyday situations may find this course will provide practical and relevant ways to help refine and augment their own knowledge of science and mathematics. Students in this course should challenge themselves to use their hands and imaginations to make robots do something.

This course fulfills the physical science requirement.

Open to: Sophomores, juniors, and seniors

Prerequisite: Any one of Advanced Algebraic and Geometric Analysis, Intensive Algebra 2, Functions, Trigonometry & Statistics, Algebra 2 (B+), or Intensive Geometry with Trigonometry (B-)

### Intensive Physical Computing & Robotics

This course is cross-listed in the math and science departments. This pace of this course is parallel to Physical Computing & Robotics, but there is a difference in depth. Specifically, students in the Intensive class are required to solve about 25 percent more problems and are expected to show mastery of the basic topics as well as learn additional topics such as arrays, EEPROM data storage, and communications protocols. Students in this class are required to write nearly all of their algorithms from scratch. For the final project, their robots are expected to perform sophisticated autonomous tasks incorporating multiple feedback control systems.

This course fulfills the physical science requirement.

Open to: Sophomores, juniors, and seniors

Prerequisite: Intensive Geometry with Trigonometry (A-), Intensive Algebra 2 (B), or a precalculus course (can be taken concurrently). Sophomores wishing to enroll must have earned either an A- in Chemistry or a B in Intensive Chemistry.