To Örebro University

Three iterative, after school design experiments with small groups of middle school students were conducted to investigate how students represent fractional relationships between two unknowns and whether they construct reciprocal reasoning with unknowns. Of the 22 students who participated, 9 had constructed fractions as multiples of unit fractions. Seven of these 9 students constructed reciprocal reasoning with unknowns. They did so by constructing a multiplicative relationship between a unit fraction of one unknown and the other unknown. The other 2 students did not make this construction. Instead, they represented relationships between unknowns with whole number multiplication and division, and by adding unknowns and fractional parts of un-knowns. The students who constructed reciprocal reasoning demonstrated a link between frac-tions as measures and fractions as operators, one benefit of working on rational number knowledge and algebraic reasoning together. Implications for teaching include recommendations for supporting students to construct reciprocal reasoning with unknowns.