Teaching and learning of integrated Science, Technology, Engineering and Mathematics (STEM) is a rapidly growing field of research with implications for both teachers and students. This qualitative research paper explores the lived experiences of STEM teachers in order to generate a model of engaging STEM tasks. Surveys and interviews with primary and secondary STEM teachers gathered descriptions of integrated STEM tasks that the teachers had identified as engaging for their students. The data informed a model that describes the features of engaging STEM lessons. An emergent theme of benefits for teachers when students are engaged in STEM is also described. Uptake of this model as a planning tool could improve student engagement and persistence in STEM with benefits to teacher well-being.