On Reconfiguring Tree Linkages: Trees can Lock

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an $N$-link tree can have $2^{\Omega(N)}$ equivalence classes of configurations.Comment: 16 pages, 6 figures Introduction reworked and references added, as the main open problem was recently close

Nonconvex Cases for Carpenter\u27s Rulers

We consider the carpenter\u27s ruler folding problem in the plane, i.e., finding a minimum area shape with diameter 1 that accommodates foldings of any ruler whose longest link has length 1. An upper bound of 0.614 and a lower bound of 0.476 are known for convex cases. We generalize the problem to simple nonconvex cases: in this setting we improve the upper bound to 0.583 and establish the first lower bound of 0.073. A variation is to consider rulers with at most k links. The current best convex upper bounds are 0.486 for k = 3, 4 and 0.523 for k = 5, 6. These bounds also apply to nonconvex cases. We derive a better nonconvex upper bound of 0.296 for k = 3, 4

An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry - Theory and Application

Transformational Geometry Unit

The study included the development and writing of a unit on transformational geometry which involved a holistic approach including the cognitive, psychomotor, and affective domains. This unit was taught to the eighth grade class in the Oakville School District in Oakville, Washington. The results showed support that the teaching of this unit was effective

Geometry in the Transition from Primary to Post-Primary

This article is intended as a kind of precursor to the document Geometry for Post-primary School Mathematics, part of the Mathematics Syllabus for Junior Certicate issued by the Irish National Council for Curriculum and Assessment in the context of Project Maths. Our purpose is to place that document in the context of an overview of plane geometry, touching on several important pedagogical and historical aspects, in the hope that this will prove useful for teachers.Comment: 19 page

Project explorations and student learning in geometry

The purpose of this study was to examine the structural framework of the EBRPSS 8th Grade Mathematics Comprehensive Curriculum and to compare its effectiveness to a set of project-based lessons that I created for “Unit 4: Measurement and Geometry”. Two classes participated in this study. Pretest and Posttest scores were analyzed to determine if there was a significant advantage to using my supplements. Results of the analysis revealed that there was an advantage to using the supplements, in spite of the time shortage. Hopefully, the supplements implemented in this study will serve as a model for teaching “Unit 4: Measurement and Geometry”

Teaching high school geometry with tasks and activities

Task-based learning is an instructional method in which students complete coherently-structured activities in order to meet objectives set by the educator. This thesis illustrates learning communities as the ideal environment for a task-based learning classroom. It discusses the teacher’s role in a task-based classroom. This paper also describes three examples of tasks performed in my high school geometry classroom along with my observations of students’ interactions and discussions

The Material Reasoning of Folding Paper

This paper inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the 19th century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. The permitted inferential actions when reasoning by folding are controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual shared standards of argument and wide enough to provide sufficiently many problems for a practice to form. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice that adds to our understanding of the multi-faceted nature of the epistemic force of mathematical proof. We then draw on what we have learned from our contemplations about paper folding to highlight some lessons about what a study of mathematical cultures entails

Euclidian Geometry: Proposed Lesson Plans to Teach Throughout a One Semester Course

Overview We provide several engaging lesson plans that would aid in the teaching of geometry, specifically targeting Euclidian Geometry, towards students of high school age. The audience of this piece would be high school or college students who have not yet had an introduction to geometry, but have completed the standard mathematical courses leading up to this point (i.e. algebra, elementary math, etc.). This being the case the lessons and concepts realized in Chapter 1 target a basic understanding of what Euclidian Geometry is and the subsequent chapters aim specifically at underlying properties of a geometry. The main source of reference for these lessons and this document is the book Foundations of Geometry Second Edition by Gerard A. Venema. These lessons are laid out as individual lessons that could be taught at any given point of a class that was dealing with the topic of the lesson at the time. These lessons are snapshots of what would be happening in a classroom and the idea is that lessons and teaching happen in-between each of the individual lessons and ideas presented here. Each chapter will begin with a summary of the main concepts and big ideas to be addressed in the chapter. I then offer the general structure of the lesson and how it could be taught. This includes what the teacher would say in the lesson and student misconceptions and questions. My hope is that this document would act as a teaching resource for teachers looking for individual lesson plans to be implemented in their own classroom during moments that they feel are appropriate. A lesson in this paper should take one class period to teach, which I have timed out at an hour. Being that most class periods are about 45 to 50 minutes this can be shortened or it could be spread out over several days as needed and appropriate. I appreciate your reading of this document and wish you a lovely day, Joseph Willer