Pythagoras Unchained For instance, younger students might draw a square with side lengths of 1 unit, color half of it, and label the colored part as part 1. The area of part 1 is 1/2 square unit. Students can then be asked to color half of the remainder with a different color and label that part 2. The area of part 2 is 1/4 square unit. If the sequence of exercises is carefully structured even very young students might be led to conjecture 1/2+1/4+1/8... in some sense gets closer and closer to 1 but never gets larger than 1. Students could then discuss their own intuitive ideas of what they observe. Such exposure in early math courses could be invaluable later when the students must confront rigorous concepts of limit and infinity. It also gives students a chance to do math rather than just learn it. Algebra 1 students who have recently learned to calculate slope might be given this challenge: On a piece of dot paper like that described above, approximate a line through the origin with a slope of 2 . For each dot near the line, have them determine if the dot is above or below the line, correcting their approximate line placement as necessary. Then ask the students "If line and the paper are extended forever, what dots will lie on the line?" The students will not realize it at first but, if there is a point ( x, y ) that falls on a line through the origin, the slope of that line will be y/x; however, the slope of this line is 2 . We are sending students on a hunt for a rational representation of an irrational number. The discovery that the line will not ever hit any dot can lead in a natural way to student conversations about the subtleties of irrationality. Conclusion The English mathematician G. H. Hardy (2015) in his book A Mathematician's Apology, wrote: "A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas" (p. 84). The purpose of the original problem, in the authors' opinion, was to help mathematics teachers remember what it is like to be a mathematician and to challenge them to think of ways to help students think like mathematicians as well. The applications offered here are only possibilities. The authors are confident that there are many creative teachers who can construct their own mathematical explorations for their students. We would be grateful to anyone who does so and shares the ideas and the classroom experiences with us. Ralph Polley * polley.math@gmail.com Mathematics teacher, Houston Academy for International Studies Robert Noyce Master Teaching Fellow, Rice University Dr. Richard Parr * rparr@rice.edu Executive Director, Rice University School Mathematics Project Clinical Assistant Professor of Mathematics, Wiess School of Natural Sciences References: Kerins, B., Sinwell, B., Yong, D., Cuoco, A., & Stevens, G. (2015). Applications of algebra and geometry to the work of teaching. Providence: American Mathematical Society. Hardy, G. (2015). A mathematician's apology. Cambridge [England]: Cambridge University Press. 30 | Spring/Summer 2018 Texas Mathematics Teacher

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