# Mathematics Courses

Milton’s mathematics curriculum is designed to encourage students to develop their understanding of a rich variety of mathematical concepts, to recognize the spatial and quantitative dimensions of the world in which they live, and to appreciate the logical principles that inform those concepts.

Beginning in the 2019–2020 school year, the mathematics department began detracking Geometry. Any student taking Geometry will be in a heterogeneous classroom. Levels will remain in Algebra II, Precalculus, Calculus and Statistics, and require the permission of the department for placement in a given level. Math classes at Milton all have an expectation of depth, extension, abstraction and problem-solving. Student exploration builds connections across topics, and allows time to consider many concepts in a real-world context. Successful completion of Geometry and Algebra 2 fulfills the diploma requirement.

Students are required to have a graphing calculator (beginning in Algebra II). The department supports the TI-84.

**Algebra 1 with Geometry
**This course is designed for students who have not taken a full-year algebra course, or who need to strengthen their algebra skills. The course also helps students to learn the fundamentals of geometry. This course will use geometric and graphing software to explore the key concepts, which include: linear, quadratic, exponential, and absolute value functions and equations; parallel lines, triangles, polygons, congruent and similar figures; and circles, area and volume. Upon successful completion of this course, students will proceed to Algebra 2. Enrollment in this course is limited and is granted by permission of the department.

**Geometry
**Students come to this course with a substantial store of information about geometric relationships gained in previous coursework and through informal experiences. This course formalizes and extends their knowledge by emphasizing an axiomatic development of these relationships. Through explorations using software programs such as GeoGebra that allow the user to construct dynamic geometric models, students make conjectures about, and then investigate and prove, geometric relationships. Topics covered in this course include parallel lines, triangles, polygons, congruent and similar figures, circles, triangle trigonometry, coordinate geometry, area and volume, and a basic introduction to computer programming in Java.

**Algebra 2 (Honors, Regular, Foundations)
**This course builds upon the foundation developed in Algebra 1 and extends students’ knowledge and understanding of algebraic concepts. The course emphasizes visual and symbolic analyses of linear, quadratic and exponential functions, as well as exponents, logarithms, sequences and series, optimization, transformations and triangle trigonometry. Other topics may include introductions to data analysis, conic sections and the properties of real and complex numbers.

*(Prerequisite: Geometry)*

**Precalculus: Functions with Mathematical Modeling (Honors, Regular, Foundations)
**This course examines the structure, application and connections between polynomial, exponential, logarithmic and trigonometric functions, along with rational functions and limits. The course also considers some discrete math topics, including combinatorics, probability and an introduction to statistics. Projects will allow students to pursue particular interests and see real-world connections. Goals of this course include building critical thinking and mathematical communication skills. (Prerequisite: Algebra II)

**Calculus (Honors, Regular)
**In this course students use limits of infinite processes to develop differential and integral calculus; they then use these concepts to create mathematical models. The abstract properties of elementary functions are re-examined in light of these new techniques; problems drawn from the natural and social sciences provide opportunities to apply these new concepts.

*(Prerequisite: Precalculus.)*

**Calculus and Applied Economics (Honors)
**This class will introduce students to the essentials of single variable calculus and the principles of economics. Students will explore the central concepts of calculus: limits, derivatives, integrals and the Fundamental Theorem while emphasizing applications to economics. The course will also illuminate the central concepts of economics, particularly microeconomics. Economics is the study of the way consumers and producers interact in markets, and the economic way of thinking centers on cost-benefit analysis. The course will use the tools of calculus to model consumer and producer behavior and to analyze the social welfare effects of government policies.

*(Prerequisite: Precalculus. Students may not take this course if they have taken, or are planning on taking Microeconomics..)*

**Calculus (Accelerated)
**This course uses limits of infinite processes to study rates of change and areas under curves. We will then reexamine abstract properties of elementary functions in light of these new techniques. Problems drawn from the natural and social sciences provide opportunities to apply these concepts. Additional topics include infinite series, parametric equations, vector analysis, and an introduction to differential equations.

*(Prerequisite: Precalculus Honors and permission of the department chair.)*

**Statistics (Honors, Regular)
**Statistics is the science of collecting, organizing and interpreting data. Students in this course learn how to analyze data from existing data sources as well as data collected from student-designed surveys and experiments. Students will also learn the importance of randomization in the collection of data and critique the validity of third-party data. This cour se investigates the underpinnings of probability theory, random variables and probability distributions as the basis for inferential statistics. Finally, students will apply all of these techniques to real-world and self-designed studies. Students gain mastery using a variety of technologies, including, but not limited to: MyStatLab, StatCrunch, spreadsheets and the calculator.

*(Prerequisite: Precalculus or permission of the department chair.)*

**Advanced Statistical Methods (Honors)
**This course will begin with a brief review of the four stages of the statistical process that are learned in Statistics: producing data, exploratory data analysis, probability theory, and statistical inference. From there, students will explore more advanced statistical topics, including: linear regression, multiple regression (including inference and variable selection), logistic regression, one-way and multi-factor ANOVA, non-parametric methods, bootstrapping, and time series analysis. The learning of these concepts will be accompanied by hands-on exploration, including using the free statistical software program: R. In the second semester, students will use the concepts and analytical skills learned in the first semester to complete class-designed and student-designed research projects. Students will be encouraged to engage in cross-curricular exploration and utilize real-world data in their analysis. Students should be interested in collaborating with their peers, working on long-term projects, and grappling with serious inquiries about the world around them.

*(Prerequisite: Statistics and Calculus, which could be taken concurrently.)*

**Advanced Calculus and Mathematical Statistics (Honors)
**This course is a calculus-based introduction to mathematical statistics. The course will cover basic probability, random variables, probability distributions, the central limit theorem and statistical inference, including parameter estimation and hypothesis testing. There are three main goals of this course: to learn the language of probability, to improve statistical intuition, and to use calculus to express and prove random concepts. Set theory, limits, sequences and series, additional methods of integration, multiple integrals and elementary differential equations will be covered.

*(Prerequisite: Calculus.)*

**Multivariable Calculus
**This course will cover topics in multivariable calculus, including vectors, vector functions, partial derivatives, multiple integrals and vector calculus. Additional advanced topics may be included, at the discretion of the instructor.

*(Enrollment by permission of the department chair. With departmental permission, this course may be taken concurrently with Advanced Calculus and Mathematical Statistics (Honors) or Abstract Algebra and Group Theory.)*

**Abstract Algebra and Group Theory
**This course is a proof-oriented introduction to the study of concrete categories such as sets, groups, abelian groups, fields, and vector spaces, focusing on the morphisms (functions), sub-structures, quotients, and actions within each category. Within Group Theory, topics include Lagrange’s Theorem, Cayley’s Theorem, The Isomorphism Theorems, and possibly Sylow’s Theorems. Within Linear Algebra, the course will focus on coordinate vectors, dimension, matrix representations of linear transformations, change of basis, determinants, and possibly eigenvectors. In the spring term, Linear Algebra will be applied to the study of Differential Equations. If time permits, rings, modules and topologies may also be considered. Specific attention will be given to the interplay between categories, which may involve the study of diagrams and functors.

*(Enrollment by permission of the department chair. With departmental permission, this course may be taken concurrently with any course beyond Precalculus.)*

**Advanced Topics in Mathematics
**

*(Semester 1, Semester 2)*

This course permits students to pursue explorations in the field of mathematics at an advanced level, for students who have already studied calculus and statistics. Topics may include number theory, topology, combinatorics, field theory, game theory or graph theory. Designed to meet the needs of the students with mathematical ideas they wish to explore in depth, this course is a seminar-style exploration of a particular field. Note: When there is a need, and staffing permits, this course may be offered as a half course.

*(Prerequisite: Calculus and Statistics and permission of the department chair.)*

**Discrete Mathematics Seminar
**

*(Half course)*

Students will study introductory graph theory and combinatorics, which are the foundations for understanding a wide range of problems in probability, computer programming and discrete applied mathematics. Students will use specific motivating questions to direct topic exploration. Motivating questions include: What is the fewest number of colors necessary to color a map of the United States so that any pair of neighboring states are different colors? If a five-card poker hand is chosen at random, what is the probability of obtaining a flush? Can a knight move around a chessboard, landing on every square exactly once? These questions are limited to the use of discrete number systems (i.e., the counting numbers and the integers). Specific topics may include: Planar Graphs, Euler Cycles, Hamilton Circuits, Coloring Theorems, Trees, Permutations, Combinations and Recursion. Classwork will include numerical problems as well as introductory logical proofs. (This course may be taken concurrently with Algebra II or Precalculus only.)

**Mathematics and Art
**

*(Half course)*

*Classes I, II & III*

Art and mathematics do not intersect so much as overlap. From Penrose tiling and M.C. Escher’s work to Mandelbrot and Julia sets to Margaret Wertheim’s crocheted coral reef and Frank Gehry’s oeuvre, mathematical art and artistic mathematics both transform our world and help us make sense of it. Students will work in a range of media from digital design and 3D printing to fiber arts. In this half course, students will spend the first part of the fall providing a foundation for the work through readings, discussions, and virtual (and actual, as logistics permit) field trips, and develop an introductory project designed by individual students and the instructor at midyear. In the spring, students will work to develop independent projects incorporating the structures and concepts studied in the first semester.

**Mathematics and Social Justice
**

*(Half Course)*

*Classes I, II & III*

This course will encourage students to explore issues of equity and justice through a mathematical lens. We will discuss our intersecting identities, learn to question our assumptions, and think critically about how bias influences the presentation of information. Students will then work with the instructor to design and complete mathematical analyses of social issues that interest them. The specific topics and mathematical tools used by an individual will depend on that student’s interests and knowledge. For example, one student might use geometry to investigate gerrymandering, while another might use calculus to analyze mass incarceration.