Multiple ontologies for units of measure have been proposed within the Applied Ontology community, and all of these ontologies introduce an array of new classes based on supposed distinctions between quantities, quantity kinds, and measures. Units are combined using notions of dimensional analysis that often conflate the combination of units with algebraic operations on real numbers. In this paper we present an alternative approach that shifts the focus to the connection between the units of measure and the physical objects and processes that are being measured. One of the key features of this approach is that it makes minimal ontological commitments with respect to the TUpperWare upper ontology – the only new classes that are introduced are the classes for the units of measure. We propose correct and complete axiomatizations for combining units of measure, and the correct axiomatization of the relationship between the units of measure and the existing upper ontology.