This lesson can be found http://www.map.mathshell.org same as title of the post.
This is one of six cards students discussed within small groups today. A student stated, “this is going to be a thinking day,” as they began removing the clips to start reviewing their cards. Most students would quickly come up with an always, sometimes or never true. However, to create their own examples or counterexamples to either justify or refute the statements was a struggle for some of them. Several groups had similar statements for this particular card. It was when a student asked, “do they have to be triangles?” that a turning point came for some.
Within our share out as a whole group, a student shared examples of reducing area, same perimeter and less perimeter. A question they wondered…can you reduce the area but increase the perimeter?
I really enjoy days like this, students are giving me the information, I am their scribe and I am slowly learning to let them determine if they agree or disagree with each others’ claims. I’m not even sure where the key is, that way I am actively having to listen to their arguments to determine if I agree or not. (Shout out to Max @Math Forum, I am listening to my students, not listening for the answer!) I go through the cards myself prior to the day of the lesson, just like I require them to do. But I am still closed minded in my own thinking at times. Why would you limit the example above to only triangles? Because that is what shape was presented on the card. However, does it state triangles only? Nope.
A task like this may drive some teachers crazy. Once you start considering different shapes, you begin to see what works for one, may not work for another. I had students cutting scrap paper, tracing patty paper, measuring side lengths…without me telling them to do it.
The classic question, a square and circle have equal perimeters, which has the larger area? I will do my best to share more reflections as we wind up tomorrow, if we wind up tomorrow…depending on their questions, discussions, claims and supporting evidence.