From the National Association of Math Circles website:
"Mathematical Circles are a form of education enrichment and outreach that bring mathematicians and mathematical scientists into direct contact with pre-college students. These students, and sometimes their teachers, meet with mathematical professionals in an informal setting, after school or on weekends, to work on interesting problems or topics in mathematics. The goal is to get the students excited about the mathematics, giving them a setting that encourages them to become passionate about mathematics."
I have worked with the Math Circles of Chicago since Fall 2012, primarily writing lessons for the Haynes group (5th-6th grades), and teaching Haynes at our Lane Tech site. I have also taught high schoolers at our Pilsen site during a summer program. Below are some of the worksheets we've used with some of our lessons in the past. Some of the ideas/problems have been taken from the various math circle resources including books from the MSRI Math Circles Library series and the Los Angeles Math Circle archive.