A quick post here – I want to share something delightful that a few Geometry students did this morning. We had our last test of the winter term today and here is one of the last questions:
Prove that the points A (x, y), B ( x + 1, y + 3), C (x + 4, y + 5), and D (x + 3, y + 2) are the vertices of the parallelogram ABCD. Prove this is true by one of the following two methods:
- By showing that one pair of opposite sides are congruent and parallel.
- By showing that both pairs of opposite sides are parallel to each other.
So, I was hoping that the majority of my students would take the quick and easy option of calculating slopes rather than messing with distances. I also hoped that the coordinates having variables in them would make them slow down, be careful, and remember a touch of algebra. I grade page by page and I have graded three of the papers with this problem on it. One student said ‘We can let x and y be 0 so the coordinates are (0,0), (1,3), (4,5), and (3,2)’ I love this thinking. She avoided the worry of dealing with the variables here. It’s a little slippery to determine just how clearly she was thinking here. She might have just been dodging a bullet. One student said ‘I will first transform this parallelogram by the vector <-x, -y> and then we will have the coordinates A’ (0,0), B’ (1,3), C’ (4,5), and D’ (3,2)’ Now, it is ABUNDANTLY clear that he knew exactly what he was doing. I’m so delighted by this that I felt I should share.
This and my great AP Stats classes today made for a pretty terrific day!
Don’t you love it when their “making it easier” actually shows more thinking than just blinding following the rules? I would have totally stuck with the original coordinates because I can’t think outside the box like that! Totally impressed!