Welcome!
It is my pleasure to having you visiting my Web Site. I am a Mathematician and professor of
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: creation set of
hands-on-manipulative activities that can help to conceptualize basic ideas in
Mathematics, and in particular in Geometry.
Consider
specific features on one aspect of my recent research interests, namely,
shape-to-shape transforms among polygonal regions (in 2D) and among prisms (in
3D). Let's start to visualize how one can transform a given polygonal region,
arbitrarily picked, into a different polygonal region of equal area through (a)
dissecting the given region and (b) moving around the resulting pieces, not
necessarily all of them, and without overlapping to form a new shape of equal
area.. Similarly, consider a given prism (in 3D) and try to visualize how to
transform it into a different prism of equal volume. The main notions here are
(a) Dissection operations where there are three types of dissections of a given
polygonal region; they are named with respect to certain side (edge) of the
given region; they are parallel, perpendicular or oblique (or diagonal) dissection;
see Figure 1.
Figure 1: The three dissections are
made with respect to the base of the triangular region
What I am trying to explain here is that, in 2D, starting with an
arbitrarily selected polygonal region, there are two geometric operations
involved:
1. The first operation evolves through the use
of paper folding and cutting - this paper folding act is not random, rather it
is a resemblance of a pre-visualized image in the mind of the user. Then the
cutting along the folds follows and these actions together I call ‘Dissection
Operations’. 2. The second operation evolves through the
manipulation of the resultant pieces through translation, rotation, reflection
or a combination of some of these to
come to a precise transformational resemblance of a shape of the user interest
with no overlapping and thus the original region and the resultant region are
area equivalent. As such I call this type of action as ‘Motion Operations’.
Finally, trying these two operations in (1) & (2) to have what I call Dissection-Motion-Operations and here you have it!
Shape Transform among Shapes in 2- and 3-D
or 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 different shape (say 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 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.
Furthermore, these dynamic shape-to-shape transformations, as you see, are
not shape-rigid transformations, that is, we 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 may do, rather we 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.
Interestingly, this shape to shape transform (Dissection-Motion-Operation)
is applicable on shapes in 3-D such as prisms. You best bet in this regard is
to view a new paper (in press) to appear Dec/Jan 2011. I encourage you to chat
with ne on this. Kind Regards
Scholarly Activities
Selected Publications
ED 5110 - Quantitative Research Method
ED 4151 - Mathematics / Curriculum & Instruction J/I
ED 4244 - Mathematics / Curriculum & Instruction I/S
ED 4258/4439 - Mathematics Enrichment / Curriculum &
Instruction J/I
Thank you for visiting my web page, if you would like more
information please don't hesitate to contact me.
Home Page of Dr. Medhat Rahim
Last updated February
23, 2012, more updates to follow
Send questions or comments to Medhat.Rahim@Lakeheadu.ca
http://mrahim.lakeheadu.ca
o
M. Phil. (Master of Philosophy) Pure
Mathematics, University of London, England.
o
Ph.D. Mathematics Education, University of
Alberta, Canada.
o
Alberta Teaching Certificate, Alberta
Ministry of Education (Permanent)
My favorite link: Algezeera - News on Iraq war 2003 (Arabic)
Iraq for you - cultural news
Iraq recent event (English and Arabic)
Noam Chomsky Monthly
Review
The
British guardian News Paper
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