# The middle block of mathematics at Proof School features an enticing variety of geometry and analysis courses.

Geometry is the course in which students traditionally first encounter the notion of mathematical proof. Although this will hardly be the case for Proofniks, geometry does involve a unique blend of visual intuition, axiomatic rigor, artistic elegance, and at times seemingly diabolical difficulty. The overarching goals throughout our geometry courses are to guide students to an appreciation of this beautiful subject while honing their ability to structure thorough, well-written arguments.

This block also heralds the arrival of Differential Calculus, a much anticipated course that is often the culmination of high school math sequences. Our current plan is for students to take differential calculus in block three, followed by integral calculus in block four, which will provide a complete introduction to the subject for application in later math courses, physics, economics, or other natural sciences. We encourage students to then take a follow-up course in a subsequent year during Block 4 that will comprehensively prepare students for the BC Calculus AP Test. We anticipate that students will typically take Calculus in eighth, ninth, or tenth grade.

*-- Sam Vandervelde*

**Fundamentals of Geometry**

with Austin Shapiro

This course introduces students to standard concepts in geometry. Topics will include parallel lines, angles, triangles, circles, parallelograms, polygons and other common shapes. Students will study metric properties of geometric objects such as length, perimeter, area, and angle measure. Finally, the course will cover the fundamental notions of similarity, proportion, and right triangles.**Elements of Geometry**

with Sachi Hashimoto

This course will present the art of proof in the context of introductory Euclidean geometry. Students will develop their skill at mathematical writing as they study the topics of angles, parallel and perpendicular lines, triangle congruence, similar triangles, parallelograms, other common quadrilaterals, circles, and constructions. Emphasis will be placed on sound arguments and clearly written proofs.**Euclidean Geometry Lab**

with Sam Vandervelde

There are a wide variety of intriguing results that become accessible once the basics of Euclidean geometry have been mastered. In this course we will begin by developing essential tools for working with circles, such as inscribed angles and Power of a Point. Then we will investigate a sequence of elegant theorems, including radical axes, Napoleon triangles, the symmedian point, the Miquel points, and other topics at the discretion of the instructor. Throughout the course students will investigate diagrams using dynamic geometry software, make and prove conjectures, write and revise proofs, and build geometric intuition.**Differential Calculus**

with John DeIonno

Beginning with the concepts of limits and continuity, this course develops the notion of the instantaneous rate of change of a function, which is given by the derivative. Topics will include formulas for derivatives, product and quotient rules, related rates, optimization, and further applications.**Topology**

with Susan Durst

In topology, we study the abstract notion of a "space," a set in which some points are closer together than others. In a topological space there is no way to measure distance, but we can get a sense of how close together points are by looking at a sequence of nested "open sets." Points that remain together in the smaller open sets are closer to each other. In this class, we will be exploring the fundamental ideas and tools of point-set topology. The class will be run largely by discovery, through guided problem sets. Students will be expected to work collaboratively and present solutions to classmates. We will have multiple quizzes, two exams, and one major writing assignment during the block.