In Geometry, we’ve been trying to turn the course on its head. Recently, we’ve been working on reasoning and proof. One thing students weren’t able to prove was that the sum of the interior angles of a triangle always add up to 180 degrees. (That’s because we haven’t gone over anything involving parallel lines.)
They even blew up balloons and drew triangles with two right angles on them (using protractors and rulers).
[Note to self: students cannot tie balloons. Also, they will be found scattered around the student center later in the day if you let kids keep their balloons.]
But we said: assuming that you know that on a plane the sum of the interior angles of a triangle add up to 180 degrees, can you prove that quadrilaterals have a sum of 360 degrees for their interior angles.
And each group was able to latch onto the idea of dissection without me saying anything… breaking the quadrilateral into two triangles.
But then… then… they started to say something that scared me. They said “there are two triangles, and since each triangle is 180 degrees, the quadrilateral is 360 degrees.”
To you non-geometry teachers, this might not seem problematic. But I immediately thought: “oh gosh, these kids think of triangles and quadrilaterals and the like as having some inherent property that can be added to others. They aren’t saying the sum of the interior angles of the triangle is 180 degrees… they are saying the triangle has 180 degrees.”
So I gave them a follow-up question (which I’m proud of):
“The Blue Triangle is 180 degrees. The Pink Triangle has 180 degrees. So the Giant Triangle (the blue and pink triangles combined) must have 360 degrees. How is this possible? Did we just break math?”
One group had someone who figured it out right away, but the others took a good amount of time trying to figure out where this argument failed. I loved it because it really showed them a misconception they had.
It was the perfect question, because over the summer the other geometry teacher and I came up with the following (which we are in love with) involving triangle dissection:
Finally, to check if each group understood where these came from, we had them write a “triangle dissection expression” for the sum of the interior angles of this pentagon: