# Upper School Mathematics

Mathematics education should give all individuals the power of problem solving, the ability to participate intelligently in civic affairs, the skills needed to pursue educational and career choices, and an appreciation of the richness and beauty of mathematics along with its importance in our culture. The department strives to integrate the National Council of Teachers of Mathematics Standards into each course. All classes provide opportunities for students to organize their thinking, to reason logically, to choose critically from different problem-solving strategies, to learn and apply various technologies, to express their ideas both orally and in writing, and to work cooperatively with their peers.

##### Sequence of Courses

Thirty credits in mathematics are required for graduation, including one year each of Algebra I, Algebra II, and Geometry and one semester of Introduction to Statistics. Typically, students complete these required courses in the following sequence: Algebra I (one year); Geometry A (fall) and Algebra IIA (spring); and Algebra IIB (fall) and Geometry B (spring), Introduction to Statistics (fall). Students who need more credits upon completing these courses or desire further study in mathematics may choose from the pre-calculus, calculus, and advanced statistics offerings subject to completing the prerequisite courses for each.

## Courses in this department:

- Algebra I
- Geometry A: Foundations of Plane Geometry
- Algebra IIA: Introduction to Functions
- Algebra IIB: Functions
- Geometry B: Logic and Proof
- Introduction to Statistics
- Pre-Calculus with Trigonometry
- Calculus I
- Calculus II
- Advanced Statistics
- Big Data and Analytics
- Individualized Program in Mathematics
- M.A.T.H. - Math in Art, Technology, and History
- Mathematical Finance

## Algebra I

In this course, students develop skills and concepts so as to learn how to solve problems using variables. Learning occurs through discussion, exploration, and group work, with the teacher acting as a guide. Topics include real, rational, and irrational numbers; solving linear and quadratic equations; graphing linear and quadratic equations; systems of linear equations; factoring polynomials; properties of exponents; properties of radicals; and simplifying rational expressions. There is an emphasis on solving problems algebraically, as well as through other methods including modeling and graphing. Developing communication skills is essential to the progression of the students’ mathematical competence.

## Geometry A: Foundations of Plane Geometry

In this course, students use an inferential approach to problem solving to build a foundation in Euclidean geometry. Through investigations of parallel lines, triangles, quadrilaterals, and circles, students make conjectures about the nature of these shapes and search out evidence to support their ideas. Students use both physical representations of diagrams and dynamic geometric software to explore the relationships present in geometric figures. The distinction between conjecture and proof and the limitations of inferential reasoning are given particular emphasis. Other course topics include conditional logic, right triangle trigonometry, and the extension of geometric relationships to three-dimensional figures. *Prerequisite:* successful completion of Algebra I (or equivalent).

## Algebra IIA: Introduction to Functions

The first semester of second-year algebra emphasizes the role of algebra as the foundation for further mathematics study. It stresses the structure of algebra, the development of algebraic problem-solving skills, and the use of functions as models of real-world situations. This course concentrates on the application of linear equations through matrices, counting and probability methods, an introduction to sequences and functions, and conics. A TI-83, TI-83 Plus, TI-84, orTI-84 Plus is required. *Prerequisite*: successful completion of Algebra I (or equivalent). A grade of C or better is recommended.

## Algebra IIB: Functions

The second semester of second-year algebra emphasizes the role of algebra as the foundation for further mathematics study. It stresses the structure of algebra, the development of algebraic problem-solving skills, and the use of functions as models of real world situations. By studying functions and their graphs, students explore the characteristics of polynomial, rational, exponential and logarithmic functions including an introduction to complex numbers. A TI-83, TI-83 Plus, TI-84, or TI-84 Plus is required. *Prerequisite:* successful completion of Algebra IIA. A grade of C or better is recommended.

## Geometry B: Logic and Proof

This course is an introduction to formal reasoning, logic, and proof. Students examine how Euclid’s postulates are the basis of an axiomatic structure for plane geometry, and how variations on those postulates create foundations for alternative geometries. Various methods for illustrating deductive reasoning are used, including two-column proof, proof by contradiction, and derivation. While the emphasis for this course is on geometric reasoning, students extend their studies with investigations of algebraic reasoning and proof. Throughout the course, students construct proofs for important mathematical concepts such as the Pythagorean Theorem, the laws of sines and cosines, and the quadratic formula. *Prerequisites:* successful completion of Geometry A and Algebra IIA (or equivalents). Grades of C or better recommended.

## Introduction to Statistics

Collecting, representing, analyzing and interpreting data are activities of major importance in contemporary society. This statistics course emphasizes that learning to interpret data correctly is a means to developing increased awareness of social, political, and scientific issues. Students learn to create unbiased surveys and experiments; describe data objectively with graphs, tables, and numerical statistics; determine correlation between variables and understand the concept of a statistically significant result. A TI-83, TI-83 Plus, TI-84, or TI-84 Plus graphing calculator is required. *Prerequisite:* successful completion of Algebra IIB(or equivalent).

## Pre-Calculus with Trigonometry

This semester course begins with a heavy emphasis on trigonometry. Students explore concepts and applications from a circular function approach and then proceed to graphing trigonometric functions, using identities to transform expressions and trigonometric functions as mathematical models. Properties of elementary functions are reviewed, expanded upon, and applied to data analysis. Other topics may include the study of vectors, matrices, and conic sections. A TI-83, TI-83 Plus, TI-84, or TI-84 Plus graphing calculator is required. *Prerequisites: *successful completion of Algebra IIB and Geometry B (or equivalents). Grades of C or better recommended.

## Calculus I

This year-long course develops and applies the concepts of differential and integral calculus. An in-depth exploration of the underlying concepts is used to develop a foundation and understanding of limits, derivatives, and integrals. These concepts are then used in a variety of applications, such as optimization, related rates of change, and volumes of revolution. Students learn to interpret the calculus concepts from several frames of reference: graphically, numerically, algebraically, and verbally. Clear communication of ideas and process are emphasized. Material in this course is comparable to material on the AB CalculusAP exam. A TI-83, TI-83 Plus, Ti-84, or TI-84 Plus is required. *Prerequisite:*successful completion of Pre-Calculus (or equivalent). A grade of B- or better is recommended.

## Calculus II

This year-long course initially revisits the fundamental concepts of calculus, emphasizing rigorous analysis. It continues with a dual approach: exercising the application of course topics and visiting their logical, conceptual basis. The application side consists of short exercises and more involved projects. The logical, conceptual side introduces and explores the idea of rigorous proof. In both approaches students look at the derivative and the integral as they relate to the following topics: infinite series, non-Cartesian coordinate systems, vector-valued functions, and multivariable functions. A TI-83, TI-83 Plus, TI-84, or TI-84 Plus graphing calculator is required. *Prerequisite:* successful completion of Calculus I (or equivalent). A grade of B- or better is recommended.

## Advanced Statistics

*Elective, second semester*

This course is a continuation of Introduction to Statistics concentrating on statistical inference procedures. In the first half of this semester course, students analyze and interpret data. Topics include confidence intervals, tests of significance, and power and errors associated with tests of significance. In the second half of the course, students design and implement a data collection method for a project of their choosing. Students will then analyze and interpret the collected data, culminating in a presentation of the processes and findings. ATI-83, TI-83 Plus, TI-84, or TI-84 Plus graphing calculator is required. *Prerequisite:*successful completion of Introduction to Statistics (or equivalent).

## Big Data and Analytics

*Elective, first semester*

Have you ever wondered how websites like Netflix and Amazon know which products to suggest to you next? Or how the NBA uses past performance to inform recommendations for athletes? This course will explore questions such as these by using programming and statistics concepts to understand big data and analytics. Topics will include data visualization, program efficiency, probability and randomness, artificial intelligence, and data structures. This course will use Python as a programming language, in order to access the extensive graphing and statistical tools of the Pylab library. In addition to programming assignments, students will also complete short math, reading, and writing assignments relating to data science concepts. This course is cross-listed in both Mathematics and General Studies subject areas. *Prerequisite*: Computer Science I or permission of instructor.

## Individualized Program in Mathematics

This course is designed to meet the needs of those students whose previous math experiences do not fall neatly into our traditional sequence. Students in this class work on an individually designed course of study. The student’s course of study is contracted with the teacher for either a semester or a year, and the student works with the teacher in a one-on-one relationship. Students interested in this course must see the department head before registration. *Prerequisite:* demonstrated ability to work conscientiously and independently.

## M.A.T.H. - Math in Art, Technology, and History

*Elective, second semester*

The overall goal of this semester elective course is to equip students with a lens to see the math that is present in the everyday world. Students explore the mathematics that exist in architecture; the patterns in natural phenomena like plants, lightning bolts, and rivers; the algorithms involved in computer programming and solving the Rubik’s cube; and the patterns in art like tessellations and the Golden Ratio. M.A.T.H. students develop an ability to recognize the mathematics in literacy and language, culture and the arts, information and communication technology. Students gain the skills and attitudes that foster lifelong learning and appreciation of mathematics in the everyday world. *Prerequisite: *Concurrent with Geometry B or beyond*.*

## Mathematical Finance

*Elective, first semester*

Students use mathematical thinking and skills to build an understanding of personal finance as well as broader knowledge of the financial world. Students learn about saving for short- and long-term goals of college, homes and retirement. Students learn about investing in stocks, bonds and funds. Students study, from a mathematical perspective, taxes and the differences between progressive and regressive taxation. Each student develops and works on a mock financial portfolio for the semester and stays current on financial news and developments through blogging and online discussion. *Prerequisite: *Concurrent with Algebra IIB or beyond*.*