This article shares activities that engage students in quality peer dialogue to build up and break down networks, revealing underlying structure and equivalence. These investigations offer students three kinds of mathematical arguments: (1) proofs of impossibility; (2) proofs by induction; and (3) proofs by exhaustion. The authors describe a bridge program that was designed for rising ninth-grade students and was field tested over two summers and included multilingual learners. The learners benefited from the architecture underlying these activities as the designs interwove forms of argument with recommendations for accelerating their deep mathematical learning.