We present a project-based approach to teaching an introductory circuit analysis course. Traditional introductory courses emphasize analysis techniques at the expense of instilling an intuitive understanding of the problem and the underlying engineering principles. We propose that an introductory circuit analysis course should teach the inter-relationships of current, voltage, power, resistance, inductance, and capacitance both mathematically and intuitively, since much of practical circuit design is not linear and cannot be reduced to linear circuit analysis problems. The fundamentals of linear circuit analysis remain important for working with linearized models of devices and for developing closed-form solutions that develop intuitive understanding of simple circuits. We describe a revision to our Circuit Analysis I course that balances both the theoretical and intuitive aspects of circuit analysis. We have also introduced a hands-on design project to engage the students in the material and to unify the laboratory exercises. The course project also introduces students to the excitement and breadth of the field of electronics, including a basic understanding of practical devices such as diodes, transistors, oscillators, and amplifiers. We present evidence that this project-based approach has benefits that include increased student interest in electronics and improved student preparation for subsequent courses.