# 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, to appreciate the logical principles that inform those concepts, and to develop their skills in critical thinking, reasoning, and communication.

Leveled courses require the permission of the department for placement. Math classes at Milton all have an expectation of depth, extension, abstraction, problem-solving, and communication. Student exploration builds connections across topics, and allows time to consider many concepts in a real-world context. Successful completion of Proof & Problem Solving (or Algebra 1 with Geometry) and Algebraic Concepts fulfills the diploma requirement.

**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, 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 Algebraic Concepts.

**Proof & Problem Solving**Students come to this course with a substantial store of information about geometric and mathematical relationships gained in previous coursework and through informal experiences. This course formalizes and extends their knowledge by emphasizing an axiomatic development of these relationships, builds problem solving and mathematical writing skills, and includes introductory work in computer programming. Some of the topics covered in this course include parallel lines, triangles, polygons, congruent and similar figures, circles, triangle trigonometry, coordinate geometry, area and volume, and an introduction to computer programming in Java. Upon completion of this course, students will have embarked on the process of communicating mathematics and formal reasoning, positioning them for success in further mathematics courses.

**Algebraic Concepts (Honors, Regular, Foundations)**This course builds upon the foundation developed in Algebra 1 and Proof & Problem Solving, extending students’ knowledge and understanding of algebraic concepts, and introducing them to work with real-world mathematical models. The course includes 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 conic sections and the properties of real and complex numbers. Students will continue to develop their ability to communicate mathematically, with a more directed focus on identifying and representing mathematical ideas in equivalent yet different ways by exploring the algebraic, graphical, numerical, and verbal representations of concepts.

*(Prerequisite: Proof & Problem Solving or Algebra 1 with Geometry.)*

**Advanced Functions (Honors, Regular)**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: Algebraic Concepts*)

**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: Advanced Functions)*

**Calculus & 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: Advanced Functions. Students may not take this course if they have taken, or are planning to take Microeconomics.)*

**Calculus (Accelerated)**This is a rapid-paced course for students who thrive in fast, intensive mathematical settings. This course uses limits of infinite processes to study rates of change and areas under curves. We will 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. In this course, students complete the study of a year and a half of Calculus in one year.

*(Prerequisite: Advanced Functions 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 course 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.

*(Prerequisite: Algebraic Concepts and Advanced Functions, which may be taken concurrently.)*

**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. Throughout the year, students will conduct a variety of research projects and 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 may be taken concurrently.)*

**Advanced Calculus & Mathematical Statistics (Honors)**This course explores the deep and rich world of mathematical statistics, with an emphasis on explaining and showing how and why things work using calculus. The course covers combinatorics, probability, random variables (both discrete and continuous) and offers choice about additional ideas, including game theory, stochastic processes, inference and hypothesis tests. No prior formal statistics is required, as concepts are taught when needed (or reviewed and extended for those who have studied statistics before). There is also an emphasis on exploration and agency in project work.

*(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.)*

**Linear & Abstract Algebra**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 Abstract Algebra, 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. Specific attention will be given to the interplay between categories, which may involve the study of diagrams and functors.Linear Algebra will be applied to the study of Linear Differential Equations. Nonlinear Differential Equations may also be pursued, based upon student choice.

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

**Advanced Topics in Mathematics**

*(Semester 1, Semester 2)*

This course permits students who have already studied calculus and statistics to pursue explorations in the field of mathematics at an advanced level. 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.

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

**Discrete Mathematics Seminar**

*(Half course)*

*Classes I, II, & III*

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 Algebraic Concepts or Advanced Functions only.*)

**Mathematics & 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 origami to knot theory and fiber arts. In this half course, students will spend the first part of the year 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 & Social Justice**

*(Semester 2)*

*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.