# Mathematics

Through studying mathematics at Putney, students learn the skills necessary for academic and personal success while building deep conceptual understanding through projects, rich problems, and explorations. Our math courses cover fundamentals for college preparation while encouraging our students to explore the beauty of mathematics and its connection with other subjects.

# Advanced Topics in Mathematics

1.0 credit

Advanced Topics students study mathematical topics not typically covered in a traditional secondary school mathematics sequence. In addition to building understanding about these topics, students further their ability to write about sophisticated mathematical concepts. Students also strengthen their abilities to use technology as tools for analysis and exploration. Mathematical literacy and writing are emphasized to prepare students for advanced study at the university level. Mathematical topics vary from year to year based on the interests and backgrounds of the students. In recent years, topics have included the study of satellite motion using computer models of differential equations, the science of passwords and encryption, and multivariable calculus. Prerequisite: Calculus 2.

# Algebra 1

1.0 credit

Algebra students focus on how to represent, model, and analyze the world they live in using mathematics. Students examine the motion of objects, population growth, climate change, and games of chance. Representation is a major thrust of the course; students regularly use tables, graphs, algebraic symbols, and verbal descriptions to represent patterns and relations. Mathematical topics include probability, linear functions, quadratic functions, and exponential functions. Computer-based technology, including graphing software, spreadsheets, and programming, is used extensively as a tool for analysis and exploration.

# Algebra 2

1.0 credit

In Algebra 2, students continue learning to use mathematics to recognize, generalize, and represent patterns in our world and to make predictions based on those patterns. The main theme of the course is understanding functions and building mathematical models for input-output relationships that are ubiquitous in everyday life. Students examine projectile motion, population growth, compound interest, and common logarithmic scales such as pH to learn about different ways that variables are used in linear, quadratic, higher-degree polynomial, rational, exponential, and logarithmic functions. Students increase their fluency with graphing programs and analytical software, use basic principles of statistics to analyze real-world data, and advance their skills in trigonometry.

# Calculus 1

1.0 credit

Students in Calculus 1 learn methods for determining how a dynamical system is changing, and how to work from a description of a changing system to a complete model of the system.

Physical examples include objects moving in space, populations growing or shrinking, objects heating or cooling, and many others. The course introduces students to the basic methods of using derivatives and integrals to investigate these systems, using a conceptual understanding of limits. We emphasize a solid physical understanding of derivatives and integrals and their connection via the Fundamental Theorem of Calculus. We leverage technology extensively in this course to assist our problem solving, system visualization, and conceptual mastery. Students also develop their ability to read mathematical writing, learning skills for understanding dense and abstract text. Please note that The Putney School does not teach to the AP curriculum and that this course is not intended to prepare students for the AP exam.

# Calculus 2

1.0 credit

Students in Calculus 2 develop a solid conceptual understanding of abstract topics of the derivative and the integral. Both projects and physical models are used to enhance understanding. As a way of grounding the concept of calculating solids of revolution, students will determine the volume of irregularly shaped vases and plot the function derived from a homemade cinnamon bun. The course begins with review of basic calculus concepts then progresses to applications of the integral, integration techniques, power series expansions of functions, and possibly the beginnings of multivariable calculus. Independent reading assignments further the development of mathematical literacy emphasized in Calculus 1. The Putney School does not teach to the AP curriculum and this course is not intended to prepare students for the AP exam.

# Computer Science 1

.5 credit

Students in this one-trimester introduction to computer science learn the basics of how a computer works as well as how to write programs to make computers execute specific tasks. Writing programs is a major component of the course. This course begins with short programming tasks but culminates in larger projects. Students who already have basic programming experience will be expected to devise and complete more challenging projects suitable to their level of expertise. Based on student interest, we may cover topics such as circuitry, Boolean logic, cryptography, computer graphics, and algorithms. Prerequisite: Algebra 1.

# Computer Science 2

.5 credit

In Computer Science 2 students build on their work from Computer Science 1. During the course of this class, students will design and build projects of their choosing in any programming language they choose. The in-class discussions will expand upon object-oriented programming and will introduce the basics of programming efficiency and runtime calculations. Prerequisite: Computer Science 1. *

* Denotes classes that combine work with a teacher (1 blocks per week) with independent, student work (3 blocks per week.) These courses require motivated students and permission of instructor.

# Geometry

1.0 credit

In Geometry, students learn how the mathematical concepts of proof and logical reasoning are used to build models of relationships among geometric figures, both real and abstract. The course draws examples from the physical campus at the Putney School and incorporates elements of art and design in real-world applications. Students develop a stronger understanding of the properties of polygons, parallel lines, and circles, as well as a beginning understanding of trigonometry. Technology, including geometric modeling software and computer programming, is used extensively as are pencil and paper techniques. More broadly, students strengthen their ability to use mathematical thinking to analyze a wide variety of situations, gather information about them, manipulate that information, and move toward finding elegant solutions to problems using both creativity and logic.

# Investment and Finance 1

.5 credit

This one-trimester elective course will address a variety of investment and financial questions including: What are stocks and bonds? What is the risk-return spectrum? What is the efficient-market hypothesis? How do loans (student, mortgage, or other) affect one’s financial well-being? What are credit ratings? What tools exist in the market for investing according to one’s ethical priorities? How do these concepts affect individual investors preparing for retirement or to pay for their children’s education? How do they affect organizations like The Putney School and its board of trustees? In addition to conceptual investigations and mastering technical vocabulary, we will investigate mathematical concepts from probability (expected value and variability in returns) and functions (exponential growth and amortization). This course will use spreadsheets extensively, but previous experience with spreadsheets is not required. Prerequisite: Algebra 2.

# Investment and Finance 2

## Not Offered 18'-19'

.5 credit

Investment and Finance 2 students build on their work from Investment and Finance 1. The first part of this elective will focus on a inflation, currency exchange, and recent financial history. After which students are expected to design and implement independent projects on topics of their choice in consultation with the teacher. Prerequisite: Investment and Finance 1.*

* Denotes classes that combine work with a teacher (2 blocks per week) with independent, student work (2 blocks per week.) These courses require motivated students and permission of instructor.

# Precalculus

1.0 credit

Precalculus students expand their earlier understanding of functions to model a larger set of nonlinear situations. Students analyze how the amount of sunlight varies cyclically throughout the year, how the levels of greenhouse gasses in our atmosphere are growing, and many examples of circular motion ranging from Ferris wheels to gears and engines. Function transformations are used extensively as a method for calibrating mathematical models to given data sets. Specific mathematical content includes sinusoidal functions, exponential and logarithmic change, vectors, sequences and series, and probability and combinatorics.

# Statistics 1

.5 credit

In this one-trimester introduction to statistics, students learn a variety of mathematical methods to manage and understand variability in data. Many scientific, social, and economic contexts will be used to motivate different techniques. We begin with ways of summarizing data quantitatively and graphically and then turn to using statistical methods to draw conclusions in the presence of variability. Our primary technology will be spreadsheets, but we will use other tools as appropriate. Topics will include linear regression, confidence intervals, and hypothesis testing. Prerequisite: Algebra 2.

# Statistics 2

## Not Offered 18'-19'

.5 credit

Statistics 2 students build on their work from Statistics 1. The first part of this elective will address stochastic simulation, sampling theory, and multivariate regression. After this, students are expected to design and implement independent projects on topics of their choice in consultation with the teacher. Prerequisite: Statistics 1.*

* Denotes classes that combine work with a teacher (2 blocks per week) with independent, student work (2 blocks per week.) These courses require motivated students and permission of instructor.