A common concern is that using large language models (LLMs) lets students offload mathematical work that tasks are designed to elicit. Yet such claims often rest on outcomes (e.g., submitted solutions), rather than on evidence of students' work with LLMs. We followed 23 first-year linear algebra students as they solved three differently designed homework tasks with LLMs. Drawing on self-regulated learning and choice-affluent environments, we interpreted each of 429 prompts as a trace of a regulatory event and compared 59 chatlogs with the problem solutions the students submitted. Prompts formed five structures, casting the LLM mainly as a solver, explainer, or validator. The mix of structures shifted across the tasks, and ten of the 15 students who completed all three tasks changed their dominant structure. Most final submissions stayed close to LLM output, including those that followed exchanges in which students had explained, conjectured, and checked their own ideas. We conclude that an LLM-aided submission is not a trustworthy marker of students' offloading of the mathematical work, and that offloading needs to be examined in each regulatory space and part by part rather than as a whole.

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