Choosing effective representations for a problem and for the person solving it has benefits, including the ability or inability to solve it. We previously devised a novel framework consisting of a language to describe representations and computational methods to analyse them in terms of their formal and cognitive properties. In this paper we demonstrate the application of this framework to a variety of notations including natural languages, formal languages, and diagrams. We show how our framework, and the analysis of representations that it enables, gives us insight into how and why we can select representations which are appropriate for both the task and the user.