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Explainable question answering systems should produce not only accurate answers but also rationales that justify their reasoning and allow humans to check their work. But what sorts of rationales are useful and how can we train systems to produce them? We propose a new style of rationale for open-book question answering, called \emph{markup-and-mask}, which combines aspects of extractive and free-text explanations. In the markup phase, the passage is augmented with free-text markup that enables each sentence to stand on its own outside the discourse context. In the masking phase, a sub-span of the marked-up passage is selected. To train a system to produce markup-and-mask rationales without annotations, we leverage in-context learning. Specifically, we generate silver annotated data by sending a series of prompts to a frozen pretrained language model, which acts as a teacher. We then fine-tune a smaller student model by training on the subset of rationales that led to correct answers. The student is "honest" in the sense that it is a pipeline: the rationale acts as a bottleneck between the passage and the answer, while the "untrusted" teacher operates under no such constraints. Thus, we offer a new way to build trustworthy pipeline systems from a combination of end-task annotations and frozen pretrained language models.