I've been having a rough year. Things haven't necessarily gone the way I would have hoped in many of my classes this year. There is a disconnect between my normal teaching strategies and the way my sophomores learn. It's been a struggle, but I'm still fighting to figure it out. Today, however, was a great day.
In my sixth period geometry class, we began with a discovery activity about triangle midsegments. It's my favorite lesson of the year. I think it's because it spirals so many of the prior ideas we've learned. Anyway... I asked "How many midpoints does a triangle have?" and thus "How many midsegments does a triangle have?". I know - I know.. this is one of those crappy questions with a definite answer. Bad teaching moment. However, one of my students turned it into a definite win. He raised his hand and said infinite. Now... he is THAT kid. You know - the one that sits in the corner and waits until he can chime in with something "cool" like infinite or no solution or Illuminati. But, this kid is brilliant - just too cool for school. He went on to say that the three midsegments of the triangle form a second triangle, which also will have three midsegments, etc. Brilliant - and yes- just yes!
Student B. Says... "So tell me again why we can't find the midpoint of a line?". Other kids respond that to take a midpoint we must know where the segment begins and ends and that we don't know this about a line. So, student B says can we just use the midpoint formula and say negative infinity + positive infinity divided by 2. We then discussed that we can't treat infinities like variables and that a negative infinity and positive infinity don't necessarily reduce to zero, unless we assume equal infinities. We started talking about how some infinities could be more infinite than other infinities... awesome conversation.
Student C then says "Doesn't a line segment have infinite midpoints?". I didn't really get where he was going, so I asked him to explain. He said "You can take the midpoint of the line segment, then take the midpoint of that point". Another student chimes in "The midpoint is a point. You can't find the midpoint of a point." So, attending to precision, he re-phrases and says "Can't you take the midpoint of a line segment then take the midpoint of one half of the line segment, then take the midpoint of that, etc". I told him that what he was doing was partitioning the segment into certain ratios - halfs, fourths, eighths, sixteenths, etc.... all while the newest midpoint was approaching the first endpoint (according to how he'd explained it).
All three of these awesome conversations from this one discovery activity about triangle midsegments. Here's the link: http://wiki.mhshs.org/images/f/f8/Midsegment_of_a_triangle_theorem.pdf
Now... if we could only have these types of discussions everyday.