We compared the two processes and looked for patterns. The images depended on three variables: the radius of the fixed circle, the radius of the moving circle, and the placement of the pen. Would it be possible to derive equations for the position of the pen? If so, could we then write our own Spirograph* program? Much to our delight, we could. You may try our version here.
Even young students can appreciate the intricate beauty of the curves generated by mathematical equations, if not the math itself. Give your students time to explore this app. Have them vary the radius of each circle and the distance from the pen to the center. What connections can they make? They may not have the mathematical language or concepts to accurately describe what's happening but they will gain an appreciation for the power and complexity of math - something that could spark a life-long interest.
*Spirograph is a registered trademark of Hasbro, Inc.