We analyzed item responses provided by a nationwide sample of 990 in-service U.S. middle grades mathematics teachers to a novel survey focused on fraction arithmetic. The survey targeted four components of reasoning in terms of measured quantities: "Referent units," "Reversibility," "Partitioning and iterating," and "Appropriateness." Using the mixture Rasch model, a psychometric model that combines unidimensional scaling with classification, we detected 3 latent classes. To understand what distinguished the performance of teachers in the three classes, we analyzed their responses to each item. One main result is that the performance of teachers across the three classes was more clearly distinguished on items targeting "Reversibility" and "Partitioning" and iterating than it was on items targeting "Referent units" and "Appropriateness." A second main result is that the three classes formed a nested structure that aligned with good performance on more and more components of reasoning targeted by the survey items.