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 Computer Science
Not Currently Offered
In Advanced Topics in Computer Science students build on their work from Computer Science 2. 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 & 2. *
* 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 from the instructor.
Advanced Topics in Mathematics
0.5 - 1.0 credit Prerequisite: Calculus 2
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. The anticipated focus for 2020-2021 is the intersection of calculus, probability and statistics. Depending on staffing and enrollment, this may be offered as a 1 or 2 trimester course.
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.
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.
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.
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
Computer Science 2
In Computer Science 2 students build on their work from Computer Science 1. Students use Python to build more complex projects. The course introduces object-oriented programming including modularity, scope, and recursion. Students set up development environments including installing external libraries and using text editors to build and run Python projects. To culminate the trimester, students will apply these new skills in an independent coding project.
Students in the Data Science elective will learn a collection of computational methods to generate informative numerical and graphical summaries of large, multivariable data sets. We will use the R programming language to identify and demonstrate patterns in data sets, e.g. voting patterns in the United Nations, or trajectories in lifespan and GDP by nation over a two hundred year span. Part of the course will be dedicated to the ethics of data science, including looking at misrepresentation of findings (particularly in visualization approaches), both bias in algorithms, and data privacy breaches. Data science is intended to be complementary to the Statistics elective, and they can be taken in either order. How are they different? Statistics emphasizes the use of probability to make decisions or inferences based on a data set, e.g. decided yes or no, a given medical treatment does, or does not, have an effect. Data science is more focused on the exploratory side of data analysis - looking for patterns before we have an idea they might exist. Students will learn some basic web-scraping techniques to collect and analyze data sets that they locate on the web. This course is open to students who have completed Algebra 2 and is offered in alternating years with Statistics.
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
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.
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.
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.