# Mathematics

By offering experiences that encourage students to explore ideas, take risks, and to think for themselves, we hope to foster a love of learning and an appreciation of mathematics. We want our students to be effective problem solvers who think logically and critically, and we emphasize the importance of creativity and resilience in working through challenging, non-routine problems. When we establish skills, the emphasis is not on rote memory, but on working through logical and developmentally appropriate processes that students understand. Concepts are first introduced in a way that provides a concrete referent and that leads to greater facility when the ideas are later expanded to a more abstract realm. Our classes are designed to foster communication and collaboration. Students are encouraged to take intellectual risks, to share their ideas, to seek out alternative approaches to problem-solving, and to develop new ways of thinking.

To see how a student progresses through math over the four years, please click here. To every reasonable extent, mathematics courses are sectioned so students may progress at a rate consistent with their ability. The final decision regarding placement for individual students is left to the department. Calculators are used throughout the curriculum; graphing calculators are required for classes beginning with Algebra 1.

The minimum requirement in mathematics is the completion of departmental courses through Geometry and Algebra 2. It is expected students continue in mathematics after fulfilling this requirement. Where possible, students are strongly encouraged to enroll in mathematics courses throughout their entire Upper School careers.

### Algebra 1

Algebra 1 is a year-long course enhanced by the study of statistics and analytic geometry, which are interwoven with algebraic concepts. Topics are covered in relation to their usefulness in solving problems rather than as ends in themselves. Using the Discovering Algebra Curriculum, we focus the course on modeling. Topics include representational statistics, linear equations, expressions, functions, systems of equations, quadratic equations, exponents and logarithms, and radical equations.

Prerequisites: Pre-Algebra and the recommendation of the department.

### Geometry

Geometry is a year-long course designed to study the properties and applications of common geometric figures in two and three dimensions. Topics include points, lines, planes, parallelism, triangles, quadrilaterals, similarity, circles area, volume and an introduction to formal proofs. Additional topics in analytic geometry are included to strengthen and enhance students’ algebraic skills. This course covers most of the topics in Advanced Geometry, with less emphasis on formal proofs and constructions.

Prerequisites: Algebra 1 and the recommendation of the department.

### Extended Geometry

For students with strong mathematical backgrounds, this course provides a challenging and collaborative approach to Euclidean and coordinate geometry, including formal deductive proofs. Topics include parallelism, congruence, polygons, circles, constructions and right-triangle trigonometry.

Prerequisites: Extended Algebra 1 and the recommendation of the department.

### Honors Geometry

For very able and motivated students, this is a challenging course in Euclidean and coordinate geometry. Topics include proof, parallelism and perpendicularity, congruence, similarity, triangle centers, right triangle trigonometry, transformations, circles, and polygons. In addition to developing proficiency in these areas, the course integrates algebraic concepts and skills throughout each unit of study. The course moves quickly, and students are expected to develop depth of understanding, the willingness to experiment, and the persistence necessary to solve unfamiliar problems. Students in the course use technology to investigate properties and discover important relationships during each unit of study. Students in this course are expected to be capable of working both independently and collaboratively, to be comfortable solving unfamiliar problems, and to proactively seek teacher support when appropriate.

Prerequisites: Honors Algebra 1 and the recommendation of the department.

### Algebra 2

Students review, in greater depth, topics first studied in Algebra I and Geometry, and progress toward more advanced algebraic techniques. There is an emphasis on order of operations, the functional relationship between variables, graphing techniques (including the use of graphing calculators) and practical applications of algebraic skills. Topics include linear, quadratic, and general polynomial functions, systems of linear equations, exponential and logarithmic functions, rational functions and probability.

Prerequisites: Geometry and the recommendation of the department.

### Extended Algebra 2

This is a challenging course for advanced students whose proven algebraic fluency is leveraged in order to pursue a rigorous study of functions. Topics include linear, quadratic, and polynomial functions, systems of linear equations, exponential and logarithmic functions, rational functions, conic sections, and probability.

Prerequisites: Extended Geometry and the recommendation of the department.

### Honors Algebra 2

For exceptionally able and highly motivated students with strong mathematical backgrounds, this course provides a rigorous approach to mathematical investigation and analysis. The goals of the course are to develop mathematical maturity, independence, and innovative thinking via a strong emphasis on problem-solving and collaborative exploration. Topics include matrices and rotational transformations, trigonometric functions, logarithms and exponentials, vectors in two and three dimensions, and volumes and areas of solids. Additional topics may be included as time allows. Student understanding is assessed through quizzes, tests, and presentations.

Prerequisites: Honors Geometry and the recommendation of the department.

### Precalculus

Precalculus is a course that focuses on representing and analyzing patterns and functional relationships using a variety of strategies, tools, and appropriate technology. It is primarily concerned with deepening students’ understanding of functions and providing an extensive investigation of trigonometry. The course emphasizes a multirepresentational approach to the topics, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally; the connections among these representations are emphasized. Specifically, we thoroughly investigate linear, polynomial, rational, exponential, logarithmic, and trigonometric functions, along with their applications. Students are introduced to matrices and conic sections as well. Although facility with manipulation and computational competence are important outcomes, they are not the core of the course. Graphing calculators and other technology will be used regularly to reinforce the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results.

Prerequisites: Algebra 2 and the recommendation of the department.

### Extended Precalculus

A preparation for studies in calculus, Advanced Precalculus explores numerical, graphical and analytical approaches to real valued functions. Students will continue their exploration of important families of functions such as exponentials, logarithms, polynomials and rationals, and will investigate trigonometric functions for the first time. Special focus is given to the topics of function transformation, domain and range, and function behavior.

Prerequisites: Extended Algebra 2 and the recommendation of the department.

### Honors Precalculus

For exceptionally able and highly motivated students with strong mathematical backgrounds, this course takes a deep, rigorous approach to topics essential to prepare for calculus.The goals of the course are to develop mathematical maturity, independence, and innovative thinking via a strong emphasis on problem-solving and collaborative exploration. Topics include complex algebra, conic sections, polynomial and rational functions, discrete mathematics, and differential calculus. Other topics will be covered as time permits.

Prerequisites: Honors Algebra 2 and the recommendation of the department.

### Calculus

Calculus introduces students to the meaning and applications of derivatives. The class starts with the study of limits and average rates of change. Using physical contexts, students study the limit of an average rate of change as a ratio between two infinitesimally small quantities. They then learn to interpret a derivative as a slope and as an instantaneous rate of change. The interplay between numerical data, graphing, and algebra is emphasized. While the course covers basic algorithms such as chain, quotient, and product rules, the focus is on conceptual understanding and the development of foundational skills. Applications include maximization, the study of linear motion, and the relationship between a function and its derivatives.

Prerequisites: Precalculus and the recommendation of the department.

### Advanced Placement Calculus AB

AP Calculus AB is a full academic year of work and is comparable to a one semester calculus course in colleges and universities. Calculus AB is primarily concerned with developing the students’ understanding of the concepts of both differential and integral calculus and providing experience with its methods and applications. The course emphasizes a multi-representational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The focus of the course is neither manipulation nor memorization of an extensive taxonomy of functions, curves, theorems, or problem types. Thus, although facility with manipulation and computational competence are important outcomes, they are not the core of the course. Graphing calculators and other technology will be used regularly to reinforce the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results. Through the use of the unifying themes of derivatives, integrals, limits, approximation, and applications and modeling, the course becomes a cohesive whole rather than a collection of unrelated topics.

Prerequisites: Extended Precalculus and the recommendation of the department.

### Advanced Placement Calculus BC

AP Calculus BC follows the standards set by College Board: This course is roughly equivalent to both first and second semester college calculus courses and extends the content learned in Calculus AB to different types of equations and introduces the topic of sequences and series. It covers topics in differential and integral calculus, including concepts and skills of limits, derivatives, definite integrals, the Fundamental Theorem of Calculus, and series. The course teaches students to approach calculus concepts and problems when they are represented graphically, numerically, analytically, and verbally, and to make connections amongst these representations. Students learn how to use technology to help solve problems, experiment, interpret results, and support conclusions.

Prerequisites: Honors Precalculus and the recommendation of the department.

### Data Analysis & Probability

This year-long elective provides a project-based approach to introductory statistics with an emphasis on developing students’ ability to use data to make evidence-based claims and informed decisions. First semester topics include: the cycle of statistical investigation, data, statistical models, distribution, center, variability, comparing groups, samples and sampling, and randomization. Second semester topics include: inference, covariation, probability distributions, confidence intervals, and hypothesis testing. Upon completion, students should be able to use appropriate technology to determine and describe important characteristics of a data set, draw inferences about a population from sample data, and interpret and communicate results.

Prerequisites: Algebra 2 and the recommendation of the department.

### Advanced Placement Statistics

AP Statistics follows the standards set by College Board. This course is equivalent to a one-semester, introductory, non-calculus-based college course in statistics. It introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. There are four themes in the AP Statistics course: exploring data, sampling and experimentation, anticipating patterns, and statistical inference. Students use technology, investigations, problem solving, and writing as they build conceptual understanding.

Prerequisites: AP Statistics can follow Extended Algebra 2/Honors Algebra 2 or any level of Precalculus with the recommendation of the department.

### Advanced Topics in Mathematics

The topic for the 2019-20 school year is Multivariable Calculus. This course covers the equivalent, in content and in rigor, of a typical college multivariable calculus course. Main topics are three- and n-dimensional geometry, differentiation including the derivative as linear transformation, vector functions and differential geometry, maxima and minima, multiple integrals, line integrals, and the theorems of Green, Gauss, and Stokes. Ways of generalizing the ideas of single-variable calculus are emphasized where applicable, as is improving mathematical communication.