Computational tutoring systems, such as educational software or interactive robots, have the potential for great societal benefit. Such systems track and assess students' knowledge via inferential methods, such as the popular Bayesian Knowledge Tracing (BKT) algorithm. However, these methods do not typically draw on the affective signals that human teachers use to assess knowledge, such as indications of discomfort, engagement, or frustration. In this paper we present a novel extension to the BKT model that uses affective data, derived autonomously from video records of children playing an interactive story-telling game with a robot, to infer student knowledge of reading skills. We find that, compared to a control group of children who played the game with only a tablet, children who interacted with an embodied social robot generated stronger affective data signals of engagement and enjoyment during the interaction. We then show that incorporating this affective data into model training improves the quality of the learned knowledge inference models. These results suggest that physically embodied, affect-aware robot tutors can provide more effective and empathic educational experiences for children, and advance both algorithmic and human-centered motivations for further development of systems that tightly integrate affect understanding and complex models of inference with interactive, educational robots.