Current Search: Yiu, Paul Y. (x)
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- Title
- Euclid, the van Hiele levels, and the Geometer's Sketchpad.
- Creator
- Thompson, Elpida, Florida Atlantic University, Yiu, Paul Y.
- Abstract/Description
-
As an educator, my greatest concern is to provide my students with instruction that will raise their level of understanding in mathematics. For geometry in particular, the van Hiele Theory is a way to measure a student's level of geometric understanding. Geometry instruction that raises a student's van Hiele level can be enhanced with two important resources, the ancient text of Euclid's Elements a contemporary dynamic geometry software program like the Geometer's Sketchpad. Euclid's Elements...
Show moreAs an educator, my greatest concern is to provide my students with instruction that will raise their level of understanding in mathematics. For geometry in particular, the van Hiele Theory is a way to measure a student's level of geometric understanding. Geometry instruction that raises a student's van Hiele level can be enhanced with two important resources, the ancient text of Euclid's Elements a contemporary dynamic geometry software program like the Geometer's Sketchpad. Euclid's Elements can be read as a book of geometric constructions rather than a list of theorems neatly arranged in logical order. The Geometer's Sketchpad is a convenient and efficient tool for geometric constructions. It is only natural to incorporate these two resources in geometry instruction. The logical structure of Euclid's Elements is intimidating to most learners, but teaching and learning need not be pursued logically linearly. This thesis is an attempt to incorporate some of the important constructions in Euclid's Elements with Geometer's Sketchpad, through the design of instruction modules in geometric constructions, to help students better understand geometry, and to improve their van Hiele level of understanding of geometry.
Show less - Date Issued
- 2006
- PURL
- http://purl.flvc.org/fcla/dt/13366
- Subject Headings
- Geometry--Study and teaching (Secondary), Mathematics--Study and teaching--Technological innovations
- Format
- Document (PDF)
- Title
- Some Construction Problems Related to the Incircle of a Triangle.
- Creator
- Bell, Amy B., Yiu, Paul Y., Florida Atlantic University
- Abstract/Description
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This thesis explores several construction problems related to the incircle of a triangle. Firstly, as a generalization of a theorem of D. W. Hansen, we find two quadruples of quantities related to a triangle which have equal sums and equal sums of squares. We also study the construction problems of triangles with centroid on the incircle, and those with a specified cevian - a median, an angle bisector, or an altitude- bisected by the incircle. Detailed analysis leads to designs of animation...
Show moreThis thesis explores several construction problems related to the incircle of a triangle. Firstly, as a generalization of a theorem of D. W. Hansen, we find two quadruples of quantities related to a triangle which have equal sums and equal sums of squares. We also study the construction problems of triangles with centroid on the incircle, and those with a specified cevian - a median, an angle bisector, or an altitude- bisected by the incircle. Detailed analysis leads to designs of animation pictures using the dynamic software Geometer's Sketchpad.
Show less - Date Issued
- 2006
- PURL
- http://purl.flvc.org/fau/fd/FA00000725
- Subject Headings
- Geometry--Problems, exercises, etc, Geometry--Study and teaching (Secondary), Mathematics--Study and teaching--Technological innovations
- Format
- Document (PDF)
- Title
- The triangle of reflections.
- Creator
- Torres, Jesus, Yiu, Paul Y., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
This thesis presents some results in triangle geometry discovered using dynamic software, namely, Geometer’s Sketchpad, and confirmed with computations using Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated with the triangle of reflections T of a given triangle T, yielding interesting triangle centers and simple loci such as circles and conics. These lead to some new triangle centers with reasonably simple coordinates, and also new properties of some...
Show moreThis thesis presents some results in triangle geometry discovered using dynamic software, namely, Geometer’s Sketchpad, and confirmed with computations using Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated with the triangle of reflections T of a given triangle T, yielding interesting triangle centers and simple loci such as circles and conics. These lead to some new triangle centers with reasonably simple coordinates, and also new properties of some known, classical centers. Particularly, we show that the Parry reflection point is the common point of two triads of circles, one associated with the tangential triangle, and another with the excentral triangle. More interestingly, we show that a certain rectangular hyperbola through the vertices of T appears as the locus of the perspector of a family of triangles perspective with T, and in a different context as the locus of the orthology center of T with another family of triangles.
Show less - Date Issued
- 2014
- PURL
- http://purl.flvc.org/fau/fd/FA00004167, http://purl.flvc.org/fau/fd/FA00004167
- Subject Headings
- Geometer's Sketchpad, Geometry -- Study and teaching, Geometry, Hyperbolic, Mathematics -- Computer network resources, Problem solving
- Format
- Document (PDF)
- Title
- The Cayley-Dickson algebras.
- Creator
- Khalil, Saidah Hasan, Florida Atlantic University, Yiu, Paul Y., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
This thesis studies the various effects of the nonassociativity of the Cayley-Dickson algebras At, t>3, especially on the structure of their automorphism groups. Beginning with the problem of composition algebra structures on euclidean spaces, we shall explain the origin of the Cayley-Dickson algebras, and give a self-contained exposition on some important results on such algebras. These algebras being nonassociative, we focus on the study of the associators of the form (u,w,v) = (uw)v - u(wv...
Show moreThis thesis studies the various effects of the nonassociativity of the Cayley-Dickson algebras At, t>3, especially on the structure of their automorphism groups. Beginning with the problem of composition algebra structures on euclidean spaces, we shall explain the origin of the Cayley-Dickson algebras, and give a self-contained exposition on some important results on such algebras. These algebras being nonassociative, we focus on the study of the associators of the form (u,w,v) = (uw)v - u(wv). The first main result, that if u and v are elements in a Cayley-Dickson algebra for which (u, w, v) = 0 for all w, then u and v generate a 2-dimensional subalgebra isomorphic to C, was conjectured by P. Yiu, and proved by P. Eakin and A. Sathaye. We shall simplify the proof given by these latter authors. This is then used to give a simple proof of R. D. Schafer's theorem on derivations of Cayley-Dickson algebras, and following also Eakin and Sathaye, a proof of the conjecture by R. B. Brown on the structure of the automorphism groups of these algebras. Two simple proofs are presented for the beautiful characterization by H. Brandt that in the Cayley algebra A3 = K, conjugation by a unit element a is an automorphism if and only if a is a 6th root of unity. We shall present a geometric proof by M. A. Zorn and a purely algebraic one. The zero divisors of the Cayley-Dickson algebra A4 are also analyzed in detail.
Show less - Date Issued
- 1993
- PURL
- http://purl.flvc.org/fcla/dt/14993
- Subject Headings
- Cayley algebras
- Format
- Document (PDF)