Summary: In this paper, we engage in a thought experiment about how students might notate their reasoning for composing fractions multiplicatively (taking a fraction of a fraction and determining its size in relation to the whole). In the thought experiment we differentiate between two levels of a fraction composition scheme, which have been identified in prior research, and discuss differences in systems of notation that students might produce, depending on the level of their scheme. We situate this discussion within a framework for learning in order to articulate how developing systems of notation with students can contribute to their learning. (ERIC)