Hold Infinity of Questions In a Piece of Grid Paper

1. Hold Infinity of QuestionsIn a Piece of Grid Paper Tatiana Shubin San Jose State University tatiana.shubin@sjsu.edu 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

2. The MTC Network developed by AIM has successfully fostered a culture of creative problemsolving 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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5. Number of Squares:

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

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

10. In general:

11. 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 m2+n2 for integers m, n ≥ 0

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

14. Food River

16. Thank you!