# Mathematics

## Algebra Ib

Algebra Ib provides the opportunity for students to finish mastering fundamental algebraic topics and techniques including evaluation and simplification of algebraic expressions, solving and graphing linear equations, linear systems, operations with polynomials, radical and rational expressions, and factoring. New topics examined in Algebra Ib include exponential equations and functions, graphing and solving quadratic and rational equations, and an introduction to data analysis and descriptive statistics. Throughout the course, students will have opportunities to develop their problem-solving strategies and number sense by using multiple methods to understand abstract concepts, mathematically interpreting problems and selecting appropriate functions, and using graphical, numeric, and algebraic representations. Graphing technology (e.g., graphing calculators and Desmos) is also introduced to aid in problem solving. Prerequisite: Algebra Ia or placement. (Full year course)

## Geometry

Geometry focuses on concepts of Euclidean Geometry with opportunities for students to apply and practice their Algebra I skills. Geometric topics examined include parallel and perpendicular lines, transformations, triangle congruence and similarity, quadrilaterals, right triangle trigonometry, circles, and area and volume. The dynamic geometry software GeoGebra is used to develop students’ inductive and deductive reasoning, to explore fundamental geometric and algebraic relationships, and to aid in geometric problem-solving. In addition, students will be expected to develop patience and resilience as they solve more lengthy Application Tasks and communicate their results through formal write-ups and oral presentations. This course will have an Advanced option that will cover all of the above topics, but with greater emphasis on problem solving, deductive proofs, and independence. Prerequisite: Algebra I, Algebra Ib, or placement. (Full year course)

## Algebra II

In Algebra II, students apply new elementary functions and algebraic techniques to model and solve problems that extend their work in Algebra and Geometry. Topics examined include transforming and modeling with linear functions, complex numbers, applications using polynomial, radical, exponential, and logarithmic functions, an introduction to rational functions, basic circular trigonometry and the sine andwith cosine functions, and an introduction to probability and data analysis. In addition, students continue to refine their problem-solving abilities by engaging with Application Tasks that require independently making mathematical conjectures about patterns and relationships using technology (e.g., graphing calculators, Desmos, and GeoGebra). They are expected to communicate their results through persuasive oral presentations and formal reports that integrate written prose, presentation of collected data using tables and graphical representations, and mathematical justification. Prerequisite: Geometry or placement. (Full year course)

Advanced Algebra II will cover all of the topics of Algebra II at a greater level of depth (but not speed) and emphasis on problem solving, deductive proofs, and independence. Additional topics may be presented as time allows. Prerequisite: teacher recommendation or placement. (Full year course; Honors)

## Math Teaching Assistants

Teaching assistants are vital contributors to our classes. TAs attend class each day, help students with practice problems and resolve homework difficulties, answer questions, and grade homework. In addition, they run review and extra-help sessions. As the year progresses, TAs plan and teach full lessons. Prerequisite: Consent of department. (Full year course)

## Precalculus

Precalculus begins with a short review of the concepts of functions and their properties and is followed by a thorough study of circular and triangular trigonometry. Students study conic sections, logarithmic and exponential functions, the graphs of rational functions, the Binomial Theorem, arithmetic and geometric sequences and series, polar coordinates, 2-D vectors, polynomial graphs and functions, and parametric equations. Students will also have the opportunity to put together and use all of the graphical representations, technology, and resources that have learned in their core math classes. Prerequisite: Algebra II, or placement. (Full year course)

Advanced Precalculus will cover all of the topics of Precalculus at a greater level of depth (but not speed). In addition, Advanced Precalculus includes 3-D vectors, DeMoivre’s Theorem, and mathematical induction. Prerequisite: Advanced Algebra II, the equivalent, or Algebra II with teacher recommendation. (Full year course; Honors)

## Statistics

Students will explore descriptive statistics and obtain the tools to assess the validity of data that they are confronted with in the media and their everyday lives. Through working with professional data set and conducting their own censuses, surveys, and other experiments, students will learn how to describe, analyze and communicate the results of their analyses orally and through formal write-ups. Statistical topics examined include central tendency and variation, data displays (e.g., bar charts, histograms, box plots, line plots, scatter plots, time series graphs, and bubble charts), the normal model, and bivariate linear regression. Concepts and data analysis will be enhanced by technology, including spreadsheets and plot.ly. Then, students will look at the theory and applications of inferential statistics, as they learn to analyze variation in data by using confidence intervals and apply inferential statistical tests to professional data sets from examples in economics, education, politics, weather, and topics of their choosing. Statistical methods examined include hypothesis tests for regression, proportions, and means (including, if time allows, the one-way analysis of variance). Students will learn to use statistical software, but the emphasis will be placed on analyzing data, interpreting the statistical output, and communicating the results of analysis. Prerequisite: Algebra II. (Fall or full year course)

## Honors Statistics

Honors Statistics is a reading-intensive Honors seminar in applied statistics. We begin by examining the topics of central tendency and variation, data displays, and probability. This leads to the study of inferential statistical topics that include the concepts of statistical models and use of samples, variation, statistical measures, sampling distributions, probability theory, tests of significance, one-way and factorial analysis of variance and covariance and elementary experimental design, multiple linear regression and correlational design, and chi-square. Students will be expected to critically analyze quantitative research, evaluate the evidence on which generalizations are made, and write quantitative methods papers by analyzing a professional data set on a topic of their choosing. In addition, students will learn to code using the industry standard statistical package SAS and emphasis is placed on using SAS to perform statistical analysis of multivariate and longitudinal data. In the past, additional topics including continuous random variables, moment-generating functions, the gamma distribution, multivariate analysis of variance, and hierarchical linear modeling have been introduced as time permits to accommodate student interests. Note: To capitalize on opportunities to meaningfully engage in research, during some years Honors Statistics is conducted as a research practicum where students take on the role of student-researchers and learn the above topics by immersing and participating in all aspects of a professional research study that culminates in professional conference presentations and co-authoring papers on the research in the spring. Experiential learning opportunities are a critical part of learning in this type of professional design practice setting and students enrolling during these years should be shared for occasional obligations and travel during and outside of school hours. Prerequisite: Precalculus, Adv Precalculus, consent of the department chair. Strong reading and writing skills. (Full year course; Honors)

## Calculus

Calculus will introduce students to the basics of differential and integral calculus. Concepts of the derivative as a slope and the integral as area will be explored using real-world examples as well as from a numerical, algebraic, visual, and verbal perspective. Activities using technology (e.g., GeoGebra, Desmos, Graphing calculators, etc.) will be utilized to help students understand concepts. Introductory rules for finding derivatives and integrals will be mastered and applied. Prerequisite: Precalculus or placement. (Full year course)

## Honors Calculus I

Honors Calculus I includes the study of limits, continuity, derivatives, integrals and their applications, slope fields, and differential equations. Concepts are approached through a four-step process: Graphically, numerically, analytically, and verbally. Graphical analysis plays a major part in the development of many concepts. Students are prepared to take the Advanced Placement Calculus AB exam in May. Prerequisite: Advanced Precalculus, Precalculus with teacher recommendation, or placement. (Full year course; Honors)

## Honors Calculus II

Honors Calculus II is recommended for students with strong backgrounds in the problem-solving aspects of one-variable calculus and emphasizes the theoretical aspects of one-variable analysis. Students gain comfort in proving the key theorems and results from first year calculus, especially rigorous definitions of the various limiting processes, and understand the importance of seemingly inconsequential theorems and properties of the real numbers. In addition, students make connections between calculus and other disciplines through modeling with differential equations. Topics examined include limits and continuous mappings, the interval theorems, Darboux integrability, first order differential equations, improper integrals and the Cauchy Principal Value, techniques of integration, sequences and series, Taylor polynomials, and parametric curves and polar coordinates. Students are prepared to take the Advanced Placement Calculus BC exam in May. In the past, additional topics such as the topology and construction of the real line, multivariable methods, and metric spaces have been introduced as time permits to accommodate student interests. Prerequisite: Honors Calculus I, Calculus with teacher recommendation, consent of department chair. (Full year course; Honors)

## Game Theory

Do you play games? Do you ever wonder if you’re using “the right” strategy? What makes one strategy better than another? In this course, we explore a branch of mathematics known as game theory, which answers these questions and many more. Game theory has many applications as we face dilemmas and conflicts every day, most of which we can treat as mathematical games. We consider significant global events from fields like diplomacy, political science, anthropology, philosophy, economics, and popular culture. Specific topics include two-person zero-sum games, two person non-zero-sum games, sequential games, multiplayer games, linear optimization, and voting and power theory. (Fall or spring semester)

## Linear Algebra

In this course students learn about the algebra of vector spaces and matrices by looking at how images of objects in the plane and space are transformed in computer graphics. We do some paper-and-pencil calculations early in the course, but the computer software package Geogebra (free) will be used to do most calculations after the opening weeks. No prior experience with this software or linear algebra is necessary. Following the introduction to core concepts and skills, students analyze social networks using linear algebraic techniques. Students will learn how to model social networks using matrices and to discover things about the network with linear algebra as your tool. We will consider applications like Facebook and Google. Prerequisite: completion of Geometry and Algebra 2 or the equivalents. (Spring semester)

## Multivariable 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. The course opens with a unit on vectors, which introduces students to this critical component of advanced calculus. We then move on to study partial derivatives, double and triple integrals, and vector calculus in both two and three dimensions. Students are expected to develop fluency with vector and matrix operations. Understanding of a parametric curve as a trajectory described by a position vector is an essential concept, and this allows us to break free from 1-dimensional calculus and investigate paths, velocities, and other applications of science that exist in three-dimensional space. We study derivatives in multiple dimensions, we use the ideas of the gradient and partial derivatives to explore optimization problems with multiple variables, and we consider constrained optimization problems using Lagrangians. After our study of differentials in multiple dimensions, we move to integral calculus. We use line and surface integrals to calculate physical quantities especially relevant to mechanics and electricity and magnetism, such as work and flux, and we employ volume integrals for calculations of mass and moments of inertia. We conclude with the major theorems (Green’s, Stokes’, Gauss’) of the course applying each to some physical applications that commonly appear in calculus-based physics. Prerequisite: The equivalent of a college year of single-variable calculus, including integration techniques, such as trigonometric substitution, integration by parts, and partial fractions. Completion of the AP Calculus BC curriculum with a score of 4 or 5 on the AP Exam would be considered adequate preparation. (Full year course)

## Number Theory

Once thought of as the purest but least applicable part of mathematics, number theory is now by far the most commonly applied: every one of the millions of secure internet transmissions occurring each second is encrypted using ideas from number theory. This online high school Number Theory class covers the fundamentals of this classical, elegant, yet supremely relevant subject. This course provides a foundation for further study of number theory, but even more, it develops the skills of mathematical reasoning and proof in a concrete and intuitive way, great preparation for any future course in upper-level college mathematics or theoretical computer science. (Fall semester)