We present a design for a junior level numerical methods course that focuses on a series of five open-ended projects in applied mathematics. These projects were deliberately designed to present many of the ambiguities and complexities that appear any time we use mathematics in the real world, and so they offered the students a variety of possible approaches, each with their own advantages and disadvantages. Because these open-ended problems lacked the normal cues that tell students how to get started, they were very effective at revealing student misconceptions, and illuminating weaknesses in their understanding. Further, they were very effective at challenging the strongest students, without leaving the weaker students behind. All students were required to write up their work on each project into a formal paper. This not only taught the valuable skill of mathematical writing, but was also an important step in teaching more organized thinking skills. Over the course of the semester, the students developed impressive abilities at dealing with these sorts of realistic problems. The students became more systematic in their explorations, and more aware of the different methods that they were forced to choose from at each stage in the project. (Contains 2 figures.)