This article describes the work of outreach mathematicians introducing the topic of complex variables to eighth and ninth grade students (13- to 15-year-olds) in the US. Complex variables is an area of mathematics that is not typically studied at secondary level. The authors developed seven lessons designed to stimulate students' interest in mathematics and, through these lessons, to improve their knowledge and understanding of algebra and geometry--topics that obviously are addressed at this level. They wanted to create an environment that drew on the processes used by professional mathematicians. A key component of these lessons was the presentation of material that was unfamiliar and potentially out of reach for these students. These factors forced the students to explore new ideas and mathematical techniques. Another important component was the structuring of questions that challenged the students' mathematical approaches. This continually forced the students to refine their methods, in the same way that research mathematicians refine their techniques in response to other mathematicians' queries. (Contains 1 figure, 3 resources and 2 notes.)