We introduce a new mechanism for one-sided matching markets, inspired by procedures currently being used to match millions of students to public universities in Brazil and China. Unlike most mechanisms available in the literature, which ask students for a full preference ranking over all colleges, they are instead sequentially asked to make choices among sets of colleges. These choices are used to produce, in each step, a tentative allo-cation. If at some point it is determined that a student cannot be accepted into a college, then she is asked to make another choice among those which would tentatively accept her. Participants following the simple strategy of choosing the most preferred college in each step is a robust equilibrium that yields the Student Optimal Stable Matching. We also provide an extension in which, after running the sequential mechanism for a number of steps students are asked to submit a ranking over the colleges that are still within reach. This constitutes a novel approach to matching mechanisms. We show that the initial sequential stage clears a substantial part of the market before the rankings submission. This finding, together with empirical and simulation results, makes our proposal an attractive alternative to the sequen-tial mechanisms currently being used and the standard Gale-Shapley Deferred Acceptance mechanism for practical applications.