Hold Infinity of Questions In a Piece of Grid Paper Tatiana Shubin San Jose State University To see a world in a grain of sand, And a heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour. -- William Blake

The MTC Network developed by AIM has successfully fostered a culture of creative problem solving among middle school teachers throughout the US. MTC sessions provide teachers with opportunities to discover mathematical facts and theories, to ask and answer their own – sometimes deep and always exciting – questions.

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Number of Squares: 1 x x 236 3x325 4x416 5x59 6x64 7x71

We can choose now one of two paths to investigate: Algebra/Number theory Geometry

Try ordering them by area: How do we list tilted squares? Area 2 Area 8 What about 1, 3, 4, 5, 6, 7?

In general:

Further investigation reveals that areas 1, 2, 4, 5, 8, 9, 10, 13… are possible 3, 6, 7, 11… are impossible It’s impossible to write 3, 6, 7, 11… as m 2 +n 2 for integers m, n ≥ 0

Draw the poor fly’s path Devise a method to find a few points exactly

River Food

Thank you!