While there is little doubt that student engagement is important to academic outcomes ranging from grades to critical thinking skills and persistence, the evidence linking engagement to these academic outcomes is often mixed in higher education. To complement existing studies, our study took a multi-faceted look at engagement from behavioral, emotional, and cognitive engagement perspectives to identify differences among students in a narrow context--that of the engineering classroom. A convergent parallel mixed-methods approach used quantitative survey data collected from over 700 engineering students in seven engineering courses to identify different patterns of engagement in the classroom. Classroom observations generated qualitative data that were then analyzed to identify characteristics of classrooms that were associated with more frequent engagement. From survey-based student self-reports of five forms of engagement (attention, effort, and participation; positive and negative emotional engagement), two distinctly different patterns of engagement emerged from a k-means clustering analysis: "more engaged" and "less engaged." Task value, instructor gender, nature of and incentive for active learning, and other factors varied among the courses studied, but no single strategy for student engagement emerged as consistently successful or unsuccessful.