## Cutting Corners

What happens when you take a figure with 4 angles and cut off one angle?  You get a figure with 3 angles, right?  The girls’ answer was natural and not at all surprising.  Luckily, I had prepared some paper quadrilaterals and scissors so that they each could perform the experiment.  When they finally both succeeded at correctly counting the number of angles of the resulting figure, it was great to see the confusion and amusement in their eyes.  Each of them was holding a pentagon and not a triangle, as she predicted!

I then asked them what figure they thought would result from cutting off a corner from the pentagon.  Here the opinions varied; one thought you’d get a quadrilateral and the other a hexagon.  Once again, the scissors were at hand.  However, the results were unexpected, even for me.  While Katie ended up with a hexagon, Varya had managed to cut off not just a corner, but a whole side, and so her resulting shape was again a pentagon.  This caused much confusion, screaming, giggling,…one can perhaps imagine the scene.  However, after I managed to restore some resemblance of order, I realized that this outcome can serve as a nice segue into my next question.  “We saw that a pentagon can be cut into two figures in different ways; what are the possible combinations that can result from cutting a quadrilateral into two parts?”

Perhaps the question was too abstract, or maybe it’s just hard to find another solution when you found one, but the girls just kept getting a triangle and a pentagon over and over again (I had prepared many quadrilaterals to play with).  Finally, I told them to try cutting it in half and gave them each something close to a rectangle.  The answer of two quadrilaterals was obtained.  By this point the girls were getting tired of the activity, but I still wanted them to obtain the last possibility.  I gave them two squares and directly asked them whether they can get two triangles.  After some whining, Katie cut hers from corner to corner, thus obtaining the desired solution.

A few closing thoughts.  I had made the quadrilaterals quite small because I wanted them to be able to cut them into two with just one snip (and I had small scissors).  However, this made it harder for the girls to manipulate them and to count the angles.  I’m not sure what is the optimal solution for this.  Also, as is often with our lessons, things did not always go as I expected and it was not always clear how to proceed.  For some reason I thought that the activity would be easier for the girls than it actually was, although looking back I realize that there was not much basis for this.  Either way, I learned a lot from doing this with the girls and hopefully they learned something too :-).