Algorithms have permeated throughout civil government and society, where they are being used to make high-stakes decisions about human lives. In this paper, we first develop a cohesive framework of algorithmic decision-making adapted for the public sector (ADMAPS) that reflects the complex socio-technical interactions between \textit{human discretion}, \textit{bureaucratic processes}, and \textit{algorithmic decision-making} by synthesizing disparate bodies of work in the fields of Human-Computer Interaction (HCI), Science and Technology Studies (STS), and Public Administration (PA). We then applied the ADMAPS framework to conduct a qualitative analysis of an in-depth, eight-month ethnographic case study of the algorithms in daily use within a child-welfare agency that serves approximately 900 families and 1300 children in the mid-western United States. Overall, we found there is a need to focus on strength-based algorithmic outcomes centered in social ecological frameworks. In addition, algorithmic systems need to support existing bureaucratic processes and augment human discretion, rather than replace it. Finally, collective buy-in in algorithmic systems requires trust in the target outcomes at both the practitioner and bureaucratic levels. As a result of our study, we propose guidelines for the design of high-stakes algorithmic decision-making tools in the child-welfare system, and more generally, in the public sector. We empirically validate the theoretically derived ADMAPS framework to demonstrate how it can be useful for systematically making pragmatic decisions about the design of algorithms for the public sector.