Over the last decade, the pedagogical approach known as mathematical modeling has received increased interest in college algebra classes in the United States. Math modeling assignments ask students to develop their own problem-solving tools to address non-routine, realistic scenarios. The open-ended quality of modeling activities creates dilemmas for teachers who design the assignments and convey grading expectations to students. This article provides a critical review of our own approach to task design for modeling as well as the extensive contributions in the United Kingdom (UK) and elsewhere in Europe dating from the 1980s and 1990s. Each approach attends differently to several competing needs: modeling as a process, modeling competencies, and developing clarity in academic writing. These resources provide substantial support for instructors of college algebra and other early undergraduate mathematics classes in the U.S. who may be embarking for the first time on task design and grading in a modeling framework.