# 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.

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 and Algebraic Concepts fulfills the diploma requirement.

### Graduation Requirements

**Proof & Problem Solving**

This entry-level course aims to prepare students for the rigors and joys of the Milton mathematics curriculum by developing the art of mathematical problem-solving and proof. Students will learn to solve problems they’ve never encountered before, identify appropriate problem-solving tools, ask clear and appropriate questions, and communicate and justify their solutions. We will regularly use algebraic and geometric concepts to teach the general skills needed in high school and beyond, such as note-taking, collaborative group work, reading and writing mathematics, working with technical text, and making connections between mathematics and other disciplines or experiences. We emphasize standards of both communication and critical thinking as we provide all students with the tools they need to be successful in future mathematics courses.

**Algebraic Concepts (Honors, Regular)**

This course builds upon the foundation developed in middle school algebra 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 equations, as well as exponents, logarithms, sequences and series, optimization, transformations, and triangle trigonometry. Other topics 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.)*

### Further Study In Mathematics

**Advanced Functions (Honors)**

Honors Advanced Functions is a course about abstraction, reasoning, communication, and making connections between math topics that seem to be unrelated. Building on students’ knowledge of linear, quadratic, and exponential functions, and of geometry and trigonometry, the Honors Advanced Functions course “takes a step back” in order to extend that prior knowledge into new domains. For example, background experience with right triangles leads to work with circular trigonometric functions and quadratics, and factoring informs work with polynomial and rational functions. This course will use the concept of transformations as a unifying theme for how familiar and unfamiliar functions behave. Abstract concepts can be difficult to learn; a degree of mathematical maturity and experience may be needed for conceptual assimilation of abstractions. *(Prerequisite: Algebraic Concepts.)*

**Advanced Functions (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 includes an introduction to statistics, mathematical modeling, social justice and public health topics, and economics. 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. Students in this course will be prepared to take both Calculus and Statistics upon completion of the course. *(Prerequisite: Algebraic Concepts.)*

**Calculus (Honors)**

Honors Calculus is a course designed to cover the content of a typical college-level Calculus I course. Derivatives, limits, and integrals, along with their applications, are studied in depth. Students are frequently asked to attempt problems without having been explicitly taught how to find the solutions and to use technology to uncover core concepts and deepen their understanding of these concepts. Students also explore differential equations, motion and physics, and applications in natural sciences. Excellent algebraic, graphing, and problem-solving skills, and a solid understanding of functions/function behavior and trigonometry are assumed. *(Prerequisite: Honors Advanced Functions or Advanced Functions and permission of the department chair.)*

**Calculus (Regular)**

In this course, students will be exposed to differential and integral calculus. Topics covered in the first semester include limits and continuity, the definition of the derivative, techniques of differentiation, and related rates. The second semester focuses on integration. Topics include the definite and indefinite integral, the Fundamental Theorem of Calculus, and real-world applications. *(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 succeed when given a limited number of problems to practice. The “accelerated” descriptor in the title refers to the pace with which we move through the course, which owes to the fact there are a large number of topics to cover—we learn three semesters of math in eight months. We study calculus topics in a well-established sequence: limits, continuity, derivatives, integrals, and applications of both, including differential and parametric equations. Finally, we study infinite series. While this is not an AP course, students who take it are prepared to take the BC Calculus exam in May. *(Prerequisite: Advanced Functions Honors and permission of the department chair at the end of the placement process.)*

**Statistics (Honors, Regular)**

This class is all about data—how to gather it, how to use it properly, and how to analyze it using graphical and computational methods. Statistics—and data science more broadly—is an important area of mathematics, as almost everything in our world relies on data, and a significant number of fields, including many outside of the realm of STEM, utilize statistics and data analysis on a regular basis. The course is divided into four main areas of study: sampling and experimentation (e.g., how to conduct surveys/studies), exploring data (e.g., graphical and computational methods), anticipating patterns (e.g., probability distributions), and statistical inference (e.g., hypothesis tests and predictions). The course focuses on real-world data—local, national, and international—covering a variety of arenas from sports to finance to healthcare and more. *(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, and random variables (both discrete and continuous) and offers choices 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 foundational topics in multivariable calculus, including vector algebra, linear transformations, matrix multiplication, vector functions, and multivariable functions. We will generalize differentiation in the fall semester, covering partial derivatives, directional derivatives, the gradient, linear differentials, and the Jacobian. We will generalize integration in the spring semester, covering multiple integrals, change of variables, and the various generalizations of the Fundamental Theorem of Calculus, including Green’s Theorem, Stoke’s Theorem, and the Divergence Theorem. 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).)*

**Linear & Abstract Algebra**

This course is a proof-oriented introduction to the study of vector spaces and the field of linear algebra, as well as other concrete categories such as sets, groups, and abelian groups, depending upon time and interest. The course will cover topics within linear algebra such as coordinate vectors, dimension, matrix representations of linear transformations, change of basis, determinants, and eigenvectors. Within abstract algebra, possible topics include Lagrange’s Theorem, Cayley’s Theorem, The Isomorphism Theorems, and Sylow’s Theorems. Specific attention will be given to the interplay between categories, which may involve the study of diagrams and functors, with a focus on the morphisms, sub-structures, quotients, and actions within each category. In the spring semester, we will apply linear algebra to the solving of linear differential equations. Interested students may also pursue the study of nonlinear differential equations and/or other applications of linear (or abstract) algebra. *(Enrollment by permission of the department chair. With departmental permission, this course may be taken concurrently with Calculus.)*

### Semester and Half Course Electives

**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. This course is intended for students who have exhausted the available course offerings in the department. (Prerequisite: Calculus and Statistics, and permission of the department chair.)

**Mathematics of Elections***(Semester 1)**Classes I, II, & III*

This course will examine the mathematical basis for how elections can be run, connect voting theory to real-world implementations, and track the high-stakes 2024 U.S. elections. Students will rigorously define voting systems and draw conclusions about their properties in order to understand why there is no perfectly fair democratic system for more than two candidates. Concurrently, students will also learn about real-world implementations of voting, selection, and apportionment, such as the U.S. Electoral College, ranked-choice voting, and parliamentary systems. These theoretical and historical perspectives will be used to preview and predict the outcomes on Election Day, and then analyze the results afterward. *(This course may be taken concurrently with any class beyond Algebraic Concepts.)*

**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. *(This course may be taken concurrently with any class beyond Algebraic Concepts.)*

**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 (e.g., 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*

In this course we will consider some of the myriad connections between art and mathematics, providing students with opportunities to study concepts beyond a traditional high school scope and sequence. Students will work in a range of media to explore mathematical connections to architecture, engineering, the geometry of materials, textile creation, transporting items into space, and other topics. *(This course may be taken concurrently with any class beyond Algebraic Concepts.)*