(Bolukbasi et al., 2016) demonstrated that pretrained word embeddings can inherit gender bias from the data they were trained on. We investigate how this bias affects downstream classification tasks, using the case study of occupation classification (De-Arteaga et al., 2019). We show that traditional techniques for debiasing embeddings can actually worsen the bias of the downstream classifier by providing a less noisy channel for communicating gender information. With a relatively minor adjustment, however, we show how these same techniques can be used to simultaneously reduce bias and maintain high classification accuracy.