Learning trajectories (LTs) can inform teaching by contributing a variety of pedagogical and content-related insights and strategies. They support sequencing topic introduction and development. Because ideas evolve gradually and often rely on a careful introduction of new representations, operations, cases of numbers, structures, and definitions, LTs can help teachers avoid overwhelming students with prematurely formal concepts. They shed light on where an idea comes from and how it allows students to see certain situations differently. They support recognizing and leveraging diverse student ideas and shed light on student misconceptions and errors. Discussions about content and pedagogy often reside within grade bands (elementary, middle, and high school). In the authors' work with teachers on LTs, they have witnessed very productive exchanges across grade bands, which can strengthen vertical coherence. To illustrate this, they describe three LTs: (1) equipartitioning, (2) ratio and proportion, and (3) exponential functions in the article. These three LTs share the territory often referred to as multiplicative space (Harel and Confrey 1994; Vergnaud 1988), which includes the operations of multiplication, division, and exponentiation. Competency in multiplicative reasoning opens the door to a myriad of applications (involving ratio and proportion, percentages, density, similarity, rate, etc.).