Rotations are powerful. They can be used as a means of understanding and explaining a whole host of important results in geometry, from vertical angles, to alternate interior angles, to properties of parallelograms. In order for students to reason with rotations, the first essential task is to formulate a precise definition of rotation. The Common Core State Standards for Mathematics call for students to be able to do exactly this in Standard G-CO.4. Standard G-CO.2 requires students to be able to “describe transformations as functions that take points in the plane as inputs and give other points as outputs.” To define a rotation $R$ is to be able to describe precisely how each point in the plane responds to the action of $R$: where does it end up? Suppose that $R$ is the rotation of the plane about point $O$ through 50 degrees. Here are the key features that define $R$:

1. The center of rotation spins in place. In other words, the center is a fixed point of $R$.
• $R(O)=O$.
2. Every other point in the plane moves along a circular path through a specified number of degrees.
• $OA=OA'$
• $\angle AOA'=50^{\circ}$

That’s it! When we start to write proofs that involve rotations, we’ll see that each of the conditions above is absolutely essential. In the next few posts, I’ll give some examples of such proofs.

The task of defining rotations is also described at Illustrative Mathematics.