One of my seventh grade classes had a passionate debate last class about whether to allow curved lines in pursuing the question “How many triangles can you make with six lines?” When the dust settled, everyone was interested in the question with only straight lines are allowed, but some were also interested in the curves-allowed version. Part of their weekend homework was to put some of their line arrangements and triangle counts on Post-its that I handed out, to be compiled and organized in class. (This was my first time using Post-its as part of homework; I think it’s a keeper.)
Anyway, one student came in today excited to share an arrangement containing twenty triangles—which was more than anyone had previously found in the straight-line-only version of the problem. She counted them out at the board, with some help from her classmates. At the end, we were all sure that there were about twenty triangles, but no one was sure that there were exactly twenty. The need for a clearer exposition was both apparent and compelling.
I introduced GeoGebra later in the period, and they definitely saw the potential for using it as a tool to share their thinking—what with the ability to make triangles visible and invisible at whim. They even asked if it would do an automatic count of triangles—that’d be pretty cool. Maybe a project in the works…?
What do you think? Are there twenty triangles in her diagram?