# Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.

Notes: Sequences and Series is part of One-Variable Calculus. We felt that since this topic had so many capsules associated with it, we wanted to introduce sub-topics. Also, the Number Theory collection of capsules does not correspond to a course in Course Communities, but has topics selected by the Editorial Board for Classroom Capsules and Notes.

## Featured Items

##### The Catalan Numbers Visit the World Series

A relation between Catalan Numbers and World Series type problems

##### Proof without Words: Algebraic Areas

A visual proof of the identity
$$(a+b)^2+(a-b)^2=2(a^2+b^2)$$

##### A Geometric Approach to Linear Functions

There are three somewhat distinct topics in this article: the condition for linear functions to commute, a linear function as a transformation of the number line, and linear difference equations. A linear function $$y=f(x)=ax+b$$ can be characterized in terms of slope and the “center of reflection,” both of which reflect the geometric property of the function.

##### Generating Functions and the Electoral College

A generating function is used to count the number of ways electoral votes can be distributed.

##### A Picture for Real Arithmetic

Real arithmetic illustrated by planar stereographic projection

##### Group Operation Tables and Normalizers

Normalizers of subgroups are obtained by organizing the operation table.