The purpose of this paper is to describe an instructional sequence where students invent a method for locating lines of eigenvectors and corresponding solutions to systems of two first order linear ordinary differential equations with constant coefficients. The significance of this paper is two-fold. First, it represents an innovative alternative where the concept and calculation of eigenvectors precede those of eigenvalues. This "eigenvector first approach," which seems to more closely follow students' conceptual development, is a reversal of the typical approach where students first learn to find eigenvalues and then determine the eigenvectors. Second, we characterize the flow of this instructional sequence in terms of four types of activities: prediction, exploration, mathematization, and generalization. The approach of prediction, exploration, mathematization, and generalization represents a more general design heuristic useful for other topics in other courses. (Contains 3 figures and 7 footnotes.)