Input-output mathematics, which allows a modeler to fully consider direct and indirect relationships among conserved flows in a system, has a long history in economics with prominent use dating to Leontief in the 1930s. Nearly all previous industrial applications of input-output analysis have been grounded in the monetary flows of an economy. Here however, because of the central nature of physical flows in the environmental impact of industry, we consider physical flows to be a fundamental component of an industrial economy. Hence, we propose an input-output based approach for modeling physical flows in industry independent of their monetary implications. In this first part of a two-part article, a framework for using input-output mathematics to model material and energy flows is constructed from a foundation laid by previous research in nutrient and energy cycling in natural ecosystems. The mathematics of input-output flow analysis is presented from an ecological perspective, culminating in two core capabilities: tracing of flows with environs (investigated in this article) and characterizing system behavior with flow metrics (presented in the second article). We assert that environ analysis is an effective means for tracing flows through industrial systems while fully considering direct and indirect flow paths. We explore material flows of aluminum and five other metals in depth using environ analysis in this article.