This study investigates "of"-constructions in the predicates of two reporting verbs, "demonstrate" and "show," in academic discourse. A construction perspective is taken to examine how the two predicate constructions (["demonstrate" N1 "of" N2] and ["show" N1 "of" N2]) would differ when the information-weighting of N1 and N2 are considered. The noun phrases are compared following Sinclair's (1991) conception of semantic headedness. He notes the peculiarity of "of" through the expression of double-headed constructions (i.e., considering both N1 and N2 as the semantic heads). This study adopts this framework and applies it to analyze the "of"-constructions of the two synonymous verbs. The results show that headedness of the "of"-constructions can be used to identify the subtle differences between the two synonyms. "Demonstrate" displays greater information weight predominated by double-headed constructions and tends to be associated with abstract conception for providing evaluative (e.g., "importance," "limitation") and modal (e.g., "possibility," "ability") functions. "Show" follows closely after "demonstrate" in the double-headed constructions, but further analysis reveals that "show" varies from "demonstrate" by displaying evidential functions through its co-occurrence with nouns denoting perspectives (e.g., "pattern," "organization"), representations (e.g., "diagram," "position") and cause and effect (e.g., "results," "impact"). It was also found that only "show" displays a propensity for N2-headed nouns as characterized by N1 measure nouns (e.g., "degree," "level") and evidential nouns (e.g., "sign," "evidence"). Pedagogical implications are provided to indicate where "demonstrate" can replace "show" in academic writing.