Dr. Medhat Rahim's Home Page

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Hi there and welcome! It is nice having you visiting my Web Page. I am a professor of Mathematics and Mathematics Education here at Lakehead University, Thunder Bay, Ontario, Canada. I would like to give you a quick tour. We may start with some of my interests in Math & Math Ed. One of my focuses is creating sets of hands-on activities that help students to conceptualize basic ideas in Mathematics, and in particular in Geometry.

To give you an idea of specific features of what I have been doing over the last two decades or so, let's start with you. Imagine for the moment with me how one can succeed in transforming a given polygon, arbitrarily picked, into an area equivalent rectangle and be able to convey this shape transformation to students at around the middle school level.

Pardon me if I confuse you here! What I am trying to say here is: starting with the arbitrarily selected polygon, using paper folding, cutting, and geometric transformations (translation, rotation, reflection, or a combination of these motions), come to a precise transformational resemblance of a rectangle.
What I mean by "transformational resemblance" is the process of cutting the polygon into pieces and moving the pieces around to assemble an area equivalent rectangle, that is, the area of the given shape is invariant under such shape-transformation. This idea is not really restricted to rectangles as our end product, rather it is extendable to result into any other shape of our interest. Meanwhile the area remains invariant. For example, you may think of changing a triangle into a rectangle or into a trapezoid, pentagon, or a parallelogram. So it is really a journey-like process among polygonal shapes rather than from a given shape to a rectangle.

Furthermore, these dynamic shape-to-shape transformations, as you see, are not shape-rigid trasformations, that is, you are not taking a triangle to a triangle and a square to a square and so on as one would expect a teacher in a school would do, rather you are taking a triangle, for example, to a non-triangle or whatever you would like it to be! Mind that you are keeping the area of the original and the resulting shape the same. If you are wondering how in the world this can happen, then I would say you and I are making these two shapes piece-to-piece congruent.

I would like to refer you to a paper I published in The International Journal for Mathematical Education in Science and Technology, 17(4), 425-447.

The paper I referred you to is focused on shape-to-rectangle transformation rather than shape-to-shape transformation. Thus, it is a restricted transformation operation. Well, initially I chose to proceed this way because of the common belief that rectangles are the most essential shapes in the study of two dimensional Euclidean Geometry and are the building blocks in the development of the concept of Integration in Calculus. However, I would like to take you further into some other publications that cover the general transformation: shape-to-shape transformation. As you see, this transformation is an extention of the starting point.

I would like to refer you to a paper I published in School Science and Mathematics Journal, 2, 113-129, USA. And another one in the same journal School Science and Mathematics Journal, 3, 235-246, USA.

You may wonder who I am and what background I possess. Okay, here it is. Keep in mind that the information presented is intended to give you a clear idea as to where I am coming from.

There is a very bright era in my professional endeavours which began at the time of my graduation from the University of Alberta.  My collaboration with Professor Daiyo Sawada, a colleague, a scholar, and a friend is so dear to me that I would like to point it out  here.  Dr. Sawada and I had many memorable times working together on many research papers discussing and sometimes getting into heated arguments over "theoretical niceties",  but I always found him an understanding and thought provoking individual.  I would like to pay tribute to Daiyo, my friend and colleague, on the occasion of his early retirement from the University this year, 1998.  This tribute is not a good bye, rather it is meant to mark his bright contribution to the arena of instruction and scientific research in mathematics and mathematics education.