We probe the molecular-level behavior of the Family 1 carbohydrate-binding module (CBM) from a commonly studied fungal cellulase, the Family 7 cellobiohydrolase (Cel7A) from Trichoderma reesei, on the hydrophobic face of crystalline cellulose. With a fully atomistic model, we predict that the CBM alone exhibits regions of thermodynamic stability along a cellulose chain corresponding to a cellobiose unit, which is the catalytic product of the entire Cel7A enzyme. In addition, we determine which residues and the types of interactions that are responsible for the observed processivity length scale of the CBM: Y5, Q7, N29, and Y32. These results imply that the CBM can anchor the Cel7A enzyme at discrete points along a cellulose chain and thus aid in both recognizing cellulose chain ends for initial attachment to cellulose as well as aid in enzymatic catalysis by diffusing between stable wells on a length scale commensurate with the catalytic, processive cycle of Cel7A during cellulose hydrolysis. Comparison of other Family 1 CBMs show high functional homology to the four amino acids responsible for the processivity length scale on the surface of crystalline cellulose, which suggests that Family 1 CBMs may generally employ this type of approach for translation on the cellulose surface. Overall, this work provides further insight into the molecular-level mechanisms by which a CBM recognizes and interacts with cellulose.