On the existence of balanced chain rule task sets

<p>In mathematics education research, mathematics task sets involving <em>mixed practice</em> 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 <em>derivative computation tasks</em>, or tasks of the form “Compute <span class="math inline">\(f'(x)\)</span> of the function <span class="math inline">\(f(x)=\)</span> [elementary function].” A <em>decomposition</em> <span class="math inline">\(D\)</span> of a graph <span class="math inline">\(G=(V,E)\)</span> is a collection <span class="math inline">\(\{H_1, H_2, ..., H_t\}\)</span> of nonempty subgraphs such that <span class="math inline">\(H_i=G[E_i]\)</span> for some nonempty subset <span class="math inline">\(E_i\)</span> of <span class="math inline">\(E(G)\)</span>, and <span class="math inline">\(\{E_1, E_2, ..., E_t\}\)</span> is a partition of <span class="math inline">\(E(G)\)</span>. We extend results on decompositions of the complete directed graph due to Meszka & Skupień to construct balanced task sets that assess the Chain Rule.</p>