PYTHAGOREAN TRIPLES

1. Squares Changed to L-shapes

 ..... This is a simple activity leading to discovery of Pythagorean Triples. Start with a square on grid paper. Then rearrange the square to form a symmetric L-shape (with equal legs).    Try different squares: 2x2, 3x3, 4x4, 5x5, etc. You will see that some squares cannot be changed into symmetric L-shapes, but some can. The connection with Pythagorean Triples becomes obvious if one notices that a symmetric L-shape is geometrically the difference of two squares. This activity replaces the existing two-parameter algorithm for generating Pythagorean triples, but which has no geometric interpretation. Our activity is understandable to young children, and arrives at the same results in an engaging manner.
 MORE EXAMPLES: The examples above reveal the triples: [3,4,5] and [5,12,13]. Next, the 6x6 square gives [6,8,10], which is just [3,4,5] doubled. Next, 7x7 gives an L-shape of length 25 and width 1, which leads to the triple [7,24,25] . Next, 8x8 gives [8,15,17], and also [8,10, 6], which is a multiple of [4,5,3], a permutation of [3,4,5]. Next, 9x9 gives [9,40,41], and a triple of [3,4,5]. In this manner we generate Pythagorean triples in increasing order of the smallest member. As soon as we check all possible deletions of even-sided squares,we obtain all Pythagorean triples.