At The Renaissance International School (TRIS), math is not so much a subject to be studied as it is a field for personal exploration and discovery. Each student embarks on his or her own journey through this magical world, conquering successive challenges as they build not only their understanding of mathematical concepts, but also their creativity, a deep love of learning, and a strong sense of discipline. The following describes one of our middle school student’s math journey so far this school year.

Isaac started the semester by exploring tessellations. He began by trying to determine which regular polygons tessellate and why. In order to answer this question, Isaac found that he needed to be able to tell how many degrees there are in each of the angles of the polygon in question; Isaac was able to derive the formula ((n-2)*180)/n with ease. Isaac then moved on to an exploration of other elements in geometry, using **Euclid's Elements** as a guide. Isaac was able to quickly start working through the different propositions and became very impressed by Euclid's attention to detail. By the tenth proposition, Isaac felt comfortable enough with the structure to start trying to prove the propositions for himself.

Isaac then decided to return to math, with an eye towards quadratic equations. In order to start down that particular path, Isaac returned to his work on the factoring of polynomials (started last year). With minimal work with didactic materials, Isaac soon became confident in this work and accepted the challenge to take a chapter test from one of our reference textbooks to see how he fared. During this activity, he encountered some problems, such as those involving the difference of two squares that he wasn't sure how to solve. As a result, Isaac started focusing on the difference of two squares. Through a series of guiding questions, Isaac was able to determine that x2-y2=(x+y)(x-y). Although Isaac was able to work through these problems computationally without problem, he found it more challenging to illustrate what these equations actually mean. He finished his explorations of difference of a square by writing out a proof for his findings (including illustrations!).

At the end of the semester, Isaac started focusing on solving quadratic equations. Using his strong understanding of the factoring of polynomials, Isaac was quickly solving those quadratic equations that can be easily factored. Once he demonstrated that he was ready for the next step, Isaac was presented with a challenge: could he figure out how to solve quadratic equations that are not easily factorable? Using a combination of paper, pencil and didactic materials, Isaac got to work. A month later, through trial and error, work with the materials and some timely tips from his teacher, Isaac had finally found a formula that he insisted could be used to find x in ANY quadratic equation….yes, he had derived the quadratic formula!