# Math

- Algebra I
- Geometry
- Algebra II
- Algebra II with Trigonometry
- Advanced Algebra II with Trigonometry
- Functions, Statistics and Trigonometry (FST)
- Pre-Calculus
- Pre-Calculus/Differential Calculus
- Elements of Calculus
- Calculus I AP
- Calculus II AP
- AP Calculus BC
- Statistics AP
- Mathematics Seminar (Calculus of Several Variables)

## Algebra I

Grade 9

Required

Full Year

Prerequisites: Algebra A or Pre-Algebra

Algebra I is the foundational course of the Upper School math curriculum. It is a continuation in our algebraic sequence of courses. The focus of the course is in the development, mastery, and application of formal algebraic notation and problem solving. The topics for the course include: linear functions, quadratic functions, factoring, and radical expressions and equations. Depth of understanding, rather than breadth of material, is the driving principle of the course. Students will not only be required to solve traditional problems but also to extend their learning to include applications of algebraic thought processes.

## Geometry

Grades 9-10

Required

Full Year

Prerequisites: Algebra I

Geometry is the second course in the sequence of Upper School mathematics courses. It normally is taken the year between Algebra I and Algebra II but may be taken concurrently with Algebra II with permission of the department. The department feels that students benefit by taking Geometry between Algebra I and Algebra II because the year allows the student to mature mathematically and to put elementary algebraic skills to use before attempting the more difficult concepts and procedures introduced in Algebra II.

The course, Euclidean geometry, involves the development of a logical, deductive system through establishment of rules of argument, definitions, postulates, and theorems. The concept of deductive proof is introduced early in the course and is fully developed as the course progresses. Topics include congruent and similar figures, perpendicular and parallel lines, polygons, circles, areas, and volumes. The main emphasis of the course is on plane geometry but the work is supplemented with some aspects of solid geometry.

## Algebra II

Grades 9-11

Satisfies the Algebra II Requirement

Full year

Prerequisites: Algebra I and Geometry, or department permission

Algebra II expands the basic skills of Algebra I. The course allows the students to further develop their algebraic skills while allowing for time to practice and apply their knowledge. Much of the course is devoted to the study of functions (linear, quadratic, polynomial, exponential and logarithmic). Time is also spent simplifying expressions, factoring, solving equations of varying degrees, and graphing. Additional topics include matrices, sequences and series, and conic sections. The course is further enhanced by the use of the TI-84 graphing calculator.

## Algebra II with Trigonometry

Grades 9-11

Satisfies the Algebra II Requirement

Full year

Prerequisites: Algebra I and Geometry, or department permission

Algebra II with Trigonometry is a course designed to transition the students from Algebra I to Pre-Calculus. A large part of the course focuses in on graphing functions (linear, quadratic, absolute value, polynomial, radical, exponential and logarithmic). Throughout the year the students apply and extend their Algebra I skills through complex, multistep equation solving. The year culminates with a five-week study of trigonometry, which includes extensive study of the unit circle and applications of triangles in problem solving. A TI-84 calculator is required for the course.

## Advanced Algebra II with Trigonometry

Grades 9-11

Satisfies the Algebra II Requirement

Full year

Prerequisites: Algebra I and Geometry, AND department permission via placement test

Advanced Algebra II with Trigonometry is designed to meet the demands of exceptional math students. Topics regularly taught in Algebra II with Trigonometry as well as Pre-Calculus will be taught to mastery. Emphasis will be placed on obtaining a deep understanding of functions both algebraically and graphically. These topics include linear systems with matrix methods, polynomial functions, rational functions, exponential and logarithmic functions, conic sections, and sequences and series. Students will also begin a study of trigonometry which will continue the following year in the next course of this sequence. A graphing calculator such as the TI-84 will be used.

## Functions, Statistics and Trigonometry (FST)

Grades 11-12

Elective

Full year

Prerequisites: Algebra II or Algebra II with Trig

For seniors, FST is excellent preparation for subsequent university classes. For juniors, this course is preparation for our introductory Calculus course, the Elements of Calculus. The course will present topics from these three areas of Functions, Statistics and Trigonometry. Students will build upon their knowledge of functions gained in Algebra II while analyzing polynomial, exponential, logarithmic, and trigonometric functions. They will use these functions in applications modeling real-world situations. The study of trigonometry will include graphs, equations, and identities. Statistical concepts learned will include data analysis, normal distributions, and probability.

## Pre-Calculus

Grades 9-12

Elective

Full Year

Prerequisites: B- or higher in either Algebra II with Trigonometry or FST

Pre-Calculus is an elective course covering a variety of advanced mathematical topics. Concepts introduced in Algebra II-Trig are reinforced and extended. This course includes a thorough study of Trigonometry and its applications. Other topics studied include function analysis including exponential and logarithmic functions, analytic geometry, complex numbers and an introduction to probability and statistics with emphasis on the normal curve. Graphing calculators are used extensively. The goal of the course is to prepare students for the study of college-level courses.

## Pre-Calculus/Differential Calculus

Grades 9-11

Elective

Full Year

Prerequisites: Advanced Algebra II with Trigonometry or department permission

This course is the second year of a 3-year sequence designed to meet the demands of exceptional math students. The first semester will include an in-depth study of Trigonometry and its applications, to include polar equations and graphs, De Moivre's Theorem, and vectors. Graphing calculators, such as a TI-84, will be used extensively. During the second semester students will study limits, derivatives, and applications of the derivative. At the conclusion of the course, students will be prepared for the next course in the sequence, AP Calculus BC.

## Elements of Calculus

Grade 12

Elective

Full Year

Prerequisites: FST or Pre-Calculus

This course is designed for students who have completed a course beyond Algebra II with the goal of introducing them to the concepts of Calculus in preparation for a subsequent university-level course. Throughout the year a focus of the course will be on concepts and applications. The course will begin with a review of necessary algebraic concepts. Calculus topics will include limits, continuity, rates of change, differentiation, integration, and exponential and logarithmic functions. A graphing calculator such as a TI-84 will often be used to explore concepts and solve problems.

## Calculus I AP

Grades 9-12

Elective

Full Year

Prerequisites: B- in Pre-Calculus and department approval (Detailed info)

Calculus I is an AP course designed to enable the more capable mathematics student to earn one or two semesters of college credit by taking the AB Advanced Placement exam at the end of the year. The course begins with a brief review of Pre-Calculus topics. The concept of limit is introduced, first on an intuitive basis and then in a theoretical context. The definition of derivative and the derivative theorems are used to find the derivatives of algebraic, trigonometric, and inverse trigonometric functions. The derivative is then applied to problems involving velocity and acceleration, related rates, graphing, and maximum-minimum problems and L’Hopital’s Rule.

The indefinite integral and simple integration formulas are studied and applied to problems involving rectilinear motion of a particle. The Fundamental Theorem of Integral Calculus is developed and applied to problems of area and volume. The integration formulas for trigonometric and inverse trigonometric functions, plus differentiation and integration formulas for logarithmic and exponential functions are developed and applied. The course concludes with the study of special methods of integration. The graphing calculator is used throughout the course.

## Calculus II AP

Grades 9-12

Elective

Full Year

Prerequisites: B- or higher in Calculus I and department approval (Detailed info)

Calculus II begins with a brief review of topics covered in Calculus I, including an in-depth study of the concepts of derivative and integral. Hyperbolic functions are then introduced along with differential equations, Taylor’s Series, infinite series, polar graphs, partial differentiation, vector-valued functions, and multiple integrals. Graphing calculators are used in the course. Completion of Calculus II prepares students for the BC level of the Calculus Advanced Placement examination but is not limited to the topics needed for that examination. Depending on students’ AP scores, from one to three semesters of college credit may be earned.

## AP Calculus BC

Grades 9-12

Elective

Full-year

Prerequisites: B in Pre-Calculus/Differential Calculus and department approval (Detailed info)

AP Calculus BC is the third year of a 3-year sequence designed to meet the demands of exceptional math students. Completion of the course prepares students for the BC level of the Calculus Advanced Placement examination; topics, however, will not be limited to those needed for that examination. Depending on students’ AP scores, from one to three semesters of college credit may be earned. After a brief review of differential calculus and its applications, the Fundamental Theorem of Integral Calculus is developed and applied to problems of area and volume. Various integration methods are studied as well as some differential equations. A significant portion of the course will focus on infinite series, including Taylor’s Series. The calculus of parametric functions and polar functions will be studied. If time allows, vector-valued functions and partial differentiation, perhaps even multiple integrals, will be considered. Graphing calculators will be used throughout the course.

## Statistics AP

Grades 10-12

Elective

Full Year

Prerequisites: Department approval (Detailed info). Generally, students must earn a B- or higher in Pre-Calculus.

The main goal of this course is to prepare the students for the AP Statistics Examination in May. The focus of the course will be a more in-depth look at the application of inferential statistics as it pertains to problem solving, analysis of data, interpretation of charts and graphs, and multivariable comparisons. The use of technology, such as the TI calculator, computer programs, and the Internet will enable the student to focus more on the interpretation of data rather than its calculation. The course will enhance the students’ appreciation of statistical information and the major impact it has on our lives today.

## Mathematics Seminar (Calculus of Several Variables)

Grades 11-12

Elective

Full Year

Prerequisites: Calculus

In this course students learn to differentiate and integrate functions of several variables. We extend the Fundamental Theorem of Calculus to multiple dimensions, and the course will culminate in Green's, Stokes' and Gauss' Theorems. We will begin with a swift review of vectors, matrices, and parametric curves, with emphasis on those topics which are of value to multivariate calculus. We will then move on to study partial derivatives, double and triple integrals, and vector calculus in both two and three dimensions. Students will be expected to develop fluency with vector and matrix operations. An understanding of a parametric curve as a trajectory described by a position vector will be an essential concept, and this will allow us to break free from 1-dimensional calculus and investigate paths, velocities, and other applications of science that exist in three-dimensional space. We will study derivatives in multiple dimensions, we will use the ideas of the gradient and partial derivatives to explore optimization problems with multiple variables, and we will consider constrained optimization problems using Lagrangians. After our study of differentials in multiple dimensions, we will move to integral calculus. We will use line and surface integrals to calculate physical quantities especially relevant to mechanics and electricity and magnetism, such as work and flux, and we will employ volume integrals for calculations of mass and moments of inertia. Change of variables in multiple integration will be investigated, and we will finish with the major theorems (Green's, Stokes', Gauss') of the course, applying each to some physical applications that commonly appear in calculus-based physics.

The Upper School math curriculum provides a comprehensive progression from basic algebra skills to college-level mathematics. No matter what course level a student takes, the Math Department faculty ensures that each student learns problem-solving skills that connect to other areas of learning. Because each student learns differently, the Math Department uses various teaching methods and works at maximizing each student’s ability to master mathematics.

Offerings beyond AP Calculus II are reserved for the most outstanding mathematics students at Park Tudor. The offerings will tend to change year-to-year, depending on staffing and the interests/needs of the individual students.