This study examines secondary mathematics teachers' anticipations of student responses related to a series of cognitively demanding mathematics tasks from multiple mathematical domains presented in the context of voluntary and asynchronous online professional development modules. We analyze 283 anticipations made by 127 teachers to 17 mathematics tasks and present four distinct foci of teachers' anticipations. Teachers participating in this study focused on actions students might take when engaging with the task -- including the representations, strategies, and resources they might use -- as well as aspects of students' mathematical thinking, such as potential difficulties they might have, prior understandings they might use, and mathematical connections they might make. Teachers also attended to how students might react emotionally when presented with tasks and what actions they might take in advance or in response to their anticipations. We conclude with a discussion of ways our results can inform efforts to support improvements in mathematics teachers' practice of anticipating.