# Texas Mathematics Teacher Spring/Summer 2018 - 30

```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.

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