Yesterday’s lesson began with me drawing a rectangle, a square, and a rhombus.
Me: A rectangle is a figure with 4 sides and 4 right angles. Is a square also a rectangle?
Girls unanimously: No!
Me: Does a square have 4 sides? (Yes.) Does it have 4 right angles? (Yes.) Then why is it not a rectangle?
Varya: It doesn’t have two long sides and two short sides. (This was accompanied with drawing long and short sides in the air.)
Me: But was that in the definition of a rectangle? (Again a unanimous yes.)
We then moved on to the definition of a rhombus. I define a rhombus as a figure with 4 equal sides and ask whether a square is also a rhombus. Of course not! This time Katie volunteers the explanation, “Because its angles are not pointy.” I then take the square and rotate it by 45 degrees. “Is it now a rhombus?” Yes. “But it was not one before?” No. I decided not to insist on anything just yet. I did come back to the questions of whether a square was a rectangle or a rhombus a few more times, each time making sure to reiterate the definitions; the girls’ answers, however, did not change.
After failed attempts at classifying the figures, we started playing with their symmetries. I had prepared some cut-out figures that I folded along various lines and opened back up. We drew the rectangle’s two lines of symmetry and verified them with folding. I then asked whether they thought the rectangle had any more lines of symmetry. Katie said yes and drew the diagonals. We then folded a rectangle along the diagonal and I had the girls verify that the two parts did not land exactly on each other; we agreed that the diagonals were not lines of symmetry for a rectangle. We then proceeded to find the rhombus’ two lines of symmetry. Here again Katie initially suggested that the line that bisects opposite sides was another line of symmetry and we verified through folding that it was not. Finally, we discovered that the square has 4 lines of symmetry: two that bisect opposite sides like a rectangle’s and two that bisect opposite angles like a rhombus’.
So now the question is how to proceed. I think that the next time that I decide to come back to this topic I will just insist that a square is both a rectangle and a rhombus because it satisfies all of the conditions. I believe that the main reason why they didn’t see it this time was because of preconceived notions about the figures: everything that they had previously seen called a rectangle had one pair of sides longer than the other, and so a square must not be a rectangle. I would love to hear any other opinions that exist out there.