We examine eight cases of argumentation in relation to a proportional reasoning task--the "Paint" task--in which the "constant sum" strategy was a significant factor. Our analysis of argument follows Toulmin's (1958) approach and in the discourse we trace factors which seem to facilitate changes in argument. We find that the arguments of "constant sum strategists" develop in the presence of (i) a small group of children with conflicting strategies, (ii) a teacher-researcher who draws attention to the context of the problem as a resource for backing warrants, and (iii) a pictorial tool which can facilitate informal communication. (Contains 1 table and 3 figures.) [For complete proceedings, see ED489538.]