In this reflection of teaching, we describe a series of activities that introduce the Taylor series through dynamic visual representations with explicit connections to students' prior learning. Over the past several decades, educators have noted that curricular materials tend to present the Taylor series in a way that students often interpret as formulaic, which leads them to focus on procedural aspects. We share our instructional approach, which introduces students to graphs of polynomial functions that resemble transcendental functions. However, students are not told that they are approximating polynomials initially, and through carefully sequenced experiences, students gradually build an intuitive understanding of the Taylor series. With the goal of equipping instructors who wish to embark upon similar journeys, we describe the visual representations employed, the questions we pose to students, students' typical responses, and important topics for discussion. The activities described engage students in reasoning and sense-making throughout the learning process, which is essential for developing a robust understanding of Taylor series as well as other calculus topics.