# Mathematics

Upper School math courses emphasize the nature of mathematical thought as well as the development of facility with the applications of mathematics; students gain a deep understanding of principles before applying them. Three course sequences offer each student the best fit in terms of pace, depth, and level of abstraction. Teachers assess progress to determine the best sequence for all students, and students may switch between sequences with departmental approval. The chart below provides an overview of the three course sequences.

*Differential and Integral Calculus can be used as preparation for the AP Calculus AB Exam; Honors Differential and Integral Calculus can be used as preparation for the AP Calculus BC Exam; Adv. Prob/Stats can be used as preparation for the AP Statistics Exam.*

- Algebra I
- Intermediate Algebra
- Algebra II
- Honors Algebra II and Trigonometry
- Geometry
- Geometry and Trigonometry
- Honors Geometry
- Advanced Algebra and Trigonometry
- Precalculus
- Honors Precalculus
- Probability and Statistics
- Topics in Data Science
- Differential and Integral Calculus
- Advanced Probability and Statistics I
- Advanced Probability and Statistics II
- Honors Differential and Integral Calculus
- Honors Advanced Math Topics (Semester I)
- Honors Advanced Math Topics (Semester II)

## Algebra I

This course is appropriate for ninth grade students who have not had a full year of algebra or whose placement tests indicate a need for further study of the basic concepts of algebra. A major focus of the course is the introduction of the basic principles of the real number system, the expression of these principles in precise algebraic language and the use of these principles in algebraic proof. Other topics covered include computation with real numbers; simplification of algebraic expressions; the use of variables when solving problems; solving linear and quadratic equations; inequalities and absolute value; order in the real numbers; operations with polynomials; factoring polynomials; and operations with algebraic fractions.

*Materials: Scientific Calculator (not a graphing calculator)*

## Intermediate Algebra

This course reviews topics from Algebra I as needed before beginning a study of mathematical functions. The major topics include investigating graphing on the number plane, solving systems of equations, the use of algebraic and graphical approaches to problem solving, square root principles and their applications, the function concept, the composition of functions, the inverse of a function, linear and quadratic functions and their applications.

*Materials: Scientific Calculator (not a graphing calculator)*

## Algebra II

The central focus of this course is mathematical functions. The first semester is devoted to developing the concept of a relation as a set of ordered pairs, the language of sets, operations on sets, graphing of relations, functions as special relations, properties of functions and operations on functions. In the second semester, these ideas are used to investigate variable quantities, formulas, direct and inverse proportionality, linear functions, root and power functions, quadratic functions and their operations.

*Materials: Scientific Calculator (not a graphing calculator)*

## Honors Algebra II and Trigonometry

The focus of this course is the study of mathematical functions. The algebra portion of the course develops the concept of a function through the study and graphing of relations, the language of sets , functions as special relations, properties of functions, and operations on functions. These ideas are used to investigate variable quantities, formulas, direct and inverse proportionality, linear functions and their graphs, and quadratic functions and their graphs. The trigonometry section of the course includes unit circle and right triangle approaches to the six trigonometric functions, radian measure, graphs of the circular functions, proofs of trigonometric identities, and the use of inverse functions to solve trigonometric equations. Application problems are studied in both the algebra and trigonometry portions of the course.

*Materials: High School Mathematics Course 3, Beberman and Vaughn reprint; Trigonometry, Lial, Hornsby, Schneider, Addison-Wesley, 1997, 6th edition; Scientific Calculator (not a graphing calculator)*

## Geometry

This course approaches the study of Euclidean and coordinate geometry from both intuitive and structured perspectives. The study of logical principles, geometric transformations and constructions form the foundation for an exploration of the geometric properties of parallel lines, triangles and quadrilaterals. Congruence principles are used to prove selected properties of quadrilaterals. Unit circle trigonometry, dilations and similarity are developed using coordinate geometry approaches. Other topics include the study of polygons, angles related to circles, special right triangles, right triangle trigonometry, area and volume formulas.

*Materials: Geometry, Jurgensen, Brown, Jurgensen, Houghton Mifflin, 1994; Scientific Calculator (not a graphing calculator)*

## Geometry and Trigonometry

This course explores topics in Euclidean geometry, coordinate geometry, and trigonometry with an emphasis on problem solving and proof. Beginning with an introduction to logical principles, the deductive organization of geometry is developed through a combination of intuitive and rigorous investigations. The course is designed to give students a range of skills associated with this branch of mathematics and to continue their preparation for the study of precalculus mathematics. The first semester includes logic, isometries, geometric constructions, congruence proofs, properties of geometric figures, coordinate geometry and an introduction to unit circle trigonometry. Second semester topics include a study of dilations and similarity, definitions and graphs of the six circular functions for real number and degree measure arguments, trigonometric identities, inverse trigonometric functions, the solution of circular function equations, areas of polygons, and volumes of polyhedra.

*Materials: High School Mathematics Course 3, Beberman and Vaughn reprint; Trigonometry, Lial, Hornsby, Schneider, Addison-Wesley, 1997, 6th edition; Scientific Calculator (not a graphing calculator)*

## Honors Geometry

This course involves a study of geometric concepts using methods from Euclidean geometry, coordinate geometry, trigonometry and the theory of vectors. Euclidean proof strategies are used during the study of congruence, polygons, angles related to polygons and circles, similarity and applications of the Pythagorean Theorem. Coordinate geometry and determinants are used in the study of systems of equations, analytic geometry proofs and coordinate approaches to computing areas. Applications of trigonometry include the Laws of Sines and Cosines and the use of matrices to represent transformations of the plane. The three dimensional geometry of lines and planes is explored using vector and matrix methods.

*Materials: Geometry, Jurgensen, Brown, Jurgensen, Houghton Mifflin, 1994; Scientific Calculator (not a graphing calculator)*

## Advanced Algebra and Trigonometry

This course reviews topics from previous algebra courses as needed through the solution of linear, quadratic, and rational equations. Advanced topics will include polynomial functions, rational functions, trigonometric functions, solution of triangles, trigonometric identities, solving trigonometric equations, circular functions and their graphs, complex numbers, conic sections and sequences and series.

*Materials: Contemporary College Algebra and Trigonometry: A Graphing Approach, Hungerford, Thomson Learning, 2005; Graphing Calculator from the TI84 family*

## Precalculus

This course is a preparation for advanced mathematics courses including calculus. The major topics covered in the course include the algebra of functions, analysis of polynomial functions, complex numbers, inequalities, systems of equations, exponential and logarithmic functions, analytic geometry of the conic sections, sequences and series, and mathematical induction. Students use graphing calculators for solving equations and inequalities, solving systems of equations, and envisioning mathematical models.

*Materials: Precalculus: Mathematics for Calculus, Stewart, Thompson Learning, 6th edition; Graphing Calculator from the TI84 family*

## Honors Precalculus

This course is a preparation for advanced mathematics courses including calculus. Polynomials, polynomial functions, rational functions, polar and parametric equations are studied using analytical techniques and a graphing calculator. The study of complex numbers includes polar forms and roots of complex numbers. Other topics include order in the real number system, principles of exponents and radicals, operations on functions, exponential and logarithmic functions, proofs by mathematical induction, curve fitting and the analytic geometry of the conic sections.

*Materials: Precalculus: Mathematics for Calculus, Stewart, Thompson Learning, 6th edition; Graphing Calculator from the TI84 family*

## Probability and Statistics

This is a one-semester introduction to statistics and probability for students who wish to improve their mathematical proficiency and explore additional mathematics through applications. Topics include visual descriptions of data, numerical descriptions of data, linear regression, combinatorics, probability theory, conditional probability, probability distributions, normal distributions, confidence intervals for means and proportions, and hypothesis testing. Students will develop proficiency in the use of a graphing calculator to display data and to do statistical tests on data.

*Materials: Introductory Statistics, Neil A. Weiss, Addison-Wesley, 6th edition; Graphing Calculator from the TI-84 family*

## Topics in Data Science

Topics in Data Science is an interdisciplinary course which combines ideas and concepts of Statistics and Computer Science. In this course, the students will learn how to extract data from cyberspace (or more specifically from the internet), learn how to organize the data, and learn how to understand various aspects of the re-organized data. The main tool we will use for the course is the open-source statistical software R. Most of the first half of the course will be spent teaching aspects of R which will be used in the rest of the course. An introductory knowledge of R gained through the prerequisite course options is expected of students entering the course. All students will complete a project using a real-world data set.

*Materials: Modern Data Science with R by Benjamin S. Baumer, Daniel T. Kaplan, and Nicholas J. Horton*

## Differential and Integral Calculus

This course, which uses a college-level text, covers an advanced treatment of elementary functions, limits, continuity, derivatives, integration, calculus of the transcendental functions, and techniques of integration. The course is appropriate for students who wish to maintain proficiency in mathematics and who wish to explore additional mainstream mathematics for later work in science, social science, business or medicine. This course may be used as preparation for the Calculus AB Advanced Placement Examination.

*Materials: Calculus for AP Early Transcendentals, Rogawski, Cannon, Freeman, 2nd edition; Graphing Calculator from the TI-84 family*

## Advanced Probability and Statistics I

This course may be taken either independently of or in sequence with Advanced Probability and Statistics II. The course is designed for students who have completed a precalculus course and wish to investigate a wide variety of applications of statistical concepts. This course may be taken simultaneously with a calculus course.

Topics covered include descriptive statistics, probability, the basics of combinatorial mathematics, random variables, and experimental design. Students are introduced to R, a statistics software program that enables them to summarize data and make statistical plots.

*Materials: The Practice of Statistics, Starns, Tabor, Yates, Moore, Freeman, 5th edition; Graphing Calculator from the TI-84 family*

## Advanced Probability and Statistics II

This course extends the concepts studied in Advanced Probability and Statistics I. It may be taken simultaneously with a calculus course.

Topics covered include inference tests for proportions and means, two proportions or two means, regression, chi-squared tests, and F-tests. Students continue to use R, a statistics software program that was introduced in the first semester of the sequence. Students who take both semesters of Advanced Probability and Statistics will be prepared to take the Advanced Placement exam in Statistics.

*Materials: The Practice of Statistics, Starns, Tabor, Yates, Moore, Freeman, 5th edition; Graphing Calculator from the TI-84 family*

## Honors Differential and Integral Calculus

This course, which uses a college-level text, offers an accelerated study of a rigorous undergraduate calculus curriculum. It includes a comprehensive investigation of single-variable differentiation and integration, including several applications. Units on techniques of integration, sequences and series, and differential equations are included.

This class, with guidance from the instructor, may be used as preparation for either the Calculus AB or Calculus BC Advanced Placement exam.

*Materials: Calculus for AP Early Transcendentals, Rogawski, Cannon, Freeman, 2nd edition; Graphing Calculator from the TI-84 family*

## Honors Advanced Math Topics (Semester I)

This course may be taken either independently of or in sequence with Honors Advanced Math Topics Sem 2. Students taking this course should have a strong background in single-variable calculus.

Discrete dynamical systems are studied with an emphasis on the dynamics of the quadratic map. Included in the topics covered are formal and informal approaches to periodicity and chaos; fractals; and dynamical systems including the computer code to produce Julia Sets.

*Materials: A First Course in Chaotic Dynamical Systems, Devaney, Addison-Wesley; Graphing Calculator from the TI-84 family*

## Honors Advanced Math Topics (Semester II)

This course may be taken independently of or in sequence with Honors Advanced Math Topics Sem 1. It builds upon and extends the concepts studied in a first calculus course. Topics covered include multivariable calculus with an emphasis on parametric and polar curves in the plane, visualization of curves and surfaces in three and higher dimensions, three-dimensional vectors, differentiation in several variables, and multivariable derivatives and optimization.

Students completing this course will have the equivalent of three semesters of college calculus.

*Materials: Calculus for AP Early Transcendentals, Rogawski, Cannon, Freeman, 2nd edition; Graphing Calculator from the TI-84 family*