Pythagoras Unchained Likewise, participants drew the segments in the second column (those with an x-coordinate of 2) (see Figure 3). Figure 3. Drawn segments with an x-coordinate of 2. Again, using the Pythagorean theorem, participants started at the bottom and calculated the lengths of these segments. They were: 4, 5, 8, 13, 20, 29 . Continuing for all five columns participants produced a table of segment lengths (see Figure 4). Figure 4. Participants' exploration results for segment lengths using the Pythagorean theorem. Column Lengths 1 1 2 5 10 17 26 2 4 5 8 13 20 29 3 9 10 13 18 25 34 4 16 17 20 25 32 41 5 25 26 29 34 41 50 While Figure 4 reflects an accurate solution, the authors saw some additional patterns. Because this presentation of the problem has essentially "unchained" Pythagoras from the usual context of solving sides of a specific triangle, we thought there might be a more compact and general description, perhaps even a function which would generate all the possible segment lengths. Therefore, we generalized the problem to include all possible segment lengths. The Problem and Investigation Given an infinitely large piece of paper on which dots have been placed in a square grid pattern 1 unit apart, what segment lengths can be produced? To begin, we looked for a pattern in the table represented in Figure 4. For simplicity, we focused on the radicands (see Figure 5). We will call the list of radicands from column 1, sequence 1. We will call the list of radicands from column 2, sequence 2, and so on. Figure 5. The radicands from the participants' exploration using the Pythagorean theorem. Column Radicands 1 1 2 5 10 17 2 4 5 8 13 20 3 9 10 13 18 25 4 16 17 20 25 32 5 25 26 29 34 41 26 29 34 41 50 Pythagoras of Samos c. 570 - c. 495 BC www.txmathteachers.org Spring/Summer 2018 | 27http://www.txmathteachers.org

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