This article describes work done at a summer institute for secondary mathematics teachers. Participants in the institute were presented with problems about Pythagorean triples. The problems were sequenced in a way that allowed for a wide variety of discoveries to develop. The following observations were made by one of the participants, which the authors thought would be of interest to secondary mathematics teachers. Introduction " The challenge for teachers is to create similar experiences and opportunities for their students that lead naturally to important concepts. " Pythagoras Unchained The Rice University School Math Project (RUSMP), established in 1987 through a National Science Foundation grant, has grown into a leading mathematics education center in the state of Texas, providing support in mathematics education for districts, schools, teachers, and students. In 2016, Rice University (through RUSMP) was awarded a grant through the Robert Noyce Teacher Scholarship program. The purpose of the grant was to identify and develop 16 Houston Independent School District middle and high school teachers as Master Teacher Fellows (MTFs). As part of their professional growth, MTFs participated in the first of two 3-week summer institutes during the summer of 2017. During this summer, the MTFs investigated the problem sets in the book Applications of Algebra and Geometry to the Work of Teaching (Kerins, Sinwell, Young, Cuoco, & Stevens, 2015). Many of the problems in the book centered on an in-depth examination of the properties of Pythagorean triples. c 2 Participants were introduced to the Pythagorean triples are whole numbers a, b, and c such that a 2 + b 2 =. topic by exploring the following introductory problem: Picture a piece of graph paper. Now picture a dot at each intersection. We'll call this square dot paper. A 5-by-5 piece of square dot paper would have 5 dots in each direction - also known as a "geoboard." But the dot paper can be any size, really. We'll say the distance from a dot to its nearest neighbor is 1. Segments drawn on square dot paper must start and end at dots, but can be horizontal, vertical, or diagonal at any angle. Question: On a 6-by-6 piece of square dot paper (see Figure 1), what lengths of segments are possible? Figure 1. Dot paper provided to participants for their exploration. To begin, participants labeled one point as the origin (0,0) and started drawing segments in the first column, ( 0, 0 ) to (1, 0 ) , ( 0, 0 ) to (1,1) , ( 0, 0 ) to (1, 2 ) , ( 0, 0 ) to (1,3) , ( 0, 0 ) to (1, 4 ) , and ( 0, 0 ) to (1,5) (see Figure 2). Figure 2. Drawn segments with an x-coordinate of 1. Using the Pythagorean theorem, starting at the bottom and working upwards, participants calculated the lengths of the drawn segments. These lengths were 1, 2, 5, 10, 17, 26 . 26 | Spring/Summer 2018 Texas Mathematics Teacher

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