On the Existence of Balanced Chain Rule Task Sets

In mathematics education research, mathematics task sets involving mixed practice include tasks from many different topics within the same assignment. In this paper, we use graph decompositions to construct mixed practice task sets for Calculus I, focusing on derivative computation tasks, or tasks of the form “Compute $f^{\mathrm{\prime }}(x)$ of the function $f(x)=$ [elementary function].” A decomposition $D$ of a graph $G=(V,E)$ is a collection $\{H_1,H_2,\mathrm{.}\mathrm{.}\mathrm{.},H_t\}$ of nonempty subgraphs such that $H_i=G[E_i]$ for some nonempty subset $E_i$ of $E(G)$, and $\{E_1,E_2,\mathrm{.}\mathrm{.}\mathrm{.},E_t\}$ is a partition of $E(G)$. We extend results on decompositions of the complete directed graph due to Meszka and Skupień to construct balanced task sets that assess the Chain Rule.