Exploring Geometry With The Geometer S Sketchpad Answers

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  exploring geometry with the geometer's sketchpad answers: Geometry Turned On James King, Doris Schattschneider, 1997-10-30 Articles about the uses of active, exploratory geometry carried out with interactive computer software.
  exploring geometry with the geometer's sketchpad answers: The Geometer's Sketchpad , 1995
  exploring geometry with the geometer's sketchpad answers: Authentic Learning Activities: Geometry & Spatial Sense Brendan Kelly, 2000
  exploring geometry with the geometer's sketchpad answers: College Geometry Barbara E. Reynolds, William E. Fenton, 2012-04-13 From two authors who embrace technology in the classroom and value the role of collaborative learning comes College Geometry Using The Geometer's Sketchpad, a book that is ideal for geometry courses for both mathematics and math education majors. The book's truly discovery-based approach guides students to learn geometry through explorations of topics ranging from triangles and circles to transformational, taxicab, and hyperbolic geometries. In the process, students hone their understanding of geometry and their ability to write rigorous mathematical proofs.
  exploring geometry with the geometer's sketchpad answers: Thinking Geometrically Thomas Q. Sibley, 2015-08-14 Thinking Geometrically: A Survey of Geometries is a well written and comprehensive survey of college geometry that would serve a wide variety of courses for both mathematics majors and mathematics education majors. Great care and attention is spent on developing visual insights and geometric intuition while stressing the logical structure, historical development, and deep interconnectedness of the ideas. Students with less mathematical preparation than upper-division mathematics majors can successfully study the topics needed for the preparation of high school teachers. There is a multitude of exercises and projects in those chapters developing all aspects of geometric thinking for these students as well as for more advanced students. These chapters include Euclidean Geometry, Axiomatic Systems and Models, Analytic Geometry, Transformational Geometry, and Symmetry. Topics in the other chapters, including Non-Euclidean Geometry, Projective Geometry, Finite Geometry, Differential Geometry, and Discrete Geometry, provide a broader view of geometry. The different chapters are as independent as possible, while the text still manages to highlight the many connections between topics. The text is self-contained, including appendices with the material in Euclid’s first book and a high school axiomatic system as well as Hilbert’s axioms. Appendices give brief summaries of the parts of linear algebra and multivariable calculus needed for certain chapters. While some chapters use the language of groups, no prior experience with abstract algebra is presumed. The text will support an approach emphasizing dynamical geometry software without being tied to any particular software.
  exploring geometry with the geometer's sketchpad answers: MathFINDER Sourcebook Laurie Kreindler, Barbara Zahm, 1992 A handbook that directs teachers to existing curricula that illustrate the goals established by the Curriculum and Evaluation Standards for School Mathematics set by the National Council of Teachers of Mathematics.
  exploring geometry with the geometer's sketchpad answers: Geometric Constructions George E. Martin, 2012-12-06 Geometric constructions have been a popular part of mathematics throughout history. The first chapter here is informal and starts from scratch, introducing all the geometric constructions from high school that have been forgotten or were never learned. The second chapter formalises Plato's game, and examines problems from antiquity such as the impossibility of trisecting an arbitrary angle. After that, variations on Plato's theme are explored: using only a ruler, a compass, toothpicks, a ruler and dividers, a marked rule, or a tomahawk, ending in a chapter on geometric constructions by paperfolding. The author writes in a charming style and nicely intersperses history and philosophy within the mathematics, teaching a little geometry and a little algebra along the way. This is as much an algebra book as it is a geometry book, yet since all the algebra and geometry needed is developed within the text, very little mathematical background is required. This text has been class tested for several semesters with a master's level class for secondary teachers.
  exploring geometry with the geometer's sketchpad answers: Mathematical Models for Teaching Ann Kajander, Tom Boland, 2014-01-01 Students of mathematics learn best when taught by a teacher with a deep and conceptual understanding of the fundamentals of mathematics. In Mathematical Models for Teaching, Ann Kajander and Tom Boland argue that teachers must be equipped with a knowledge of mathematics for teaching, which is grounded in modelling, reasoning, and problem-based learning. A comprehensive exploration of models and concepts, this book promotes an understanding of the material that goes beyond memorization and recitation, which begins with effective teaching. This vital resource is divided into 15 chapters, each of which addresses a specific mathematical concept. Focusing on areas that have been identified as problematic for teachers and students, Mathematical Models for Teaching equips teachers with a different type of mathematical understanding-one that supports and encourages student development. Features: grounded in the most current research about teachers' learning contains cross-chapter connections that identify common ideas includes chapter concluding discussion questions that encourage critical thinking incorporates figures and diagrams that simplify and solidify important mathematical concepts offers further reading suggestions for instructors seeking additional information
  exploring geometry with the geometer's sketchpad answers: Discovering Mathematics , 2004
  exploring geometry with the geometer's sketchpad answers: Euclidean and Transformational Geometry Shlomo Libeskind, 2008-02-12 Ideal for mathematics majors and prospective secondary school teachers, Euclidean and Transformational Geometry provides a complete and solid presentation of Euclidean geometry with an emphasis on solving challenging problems. The author examines various strategies and heuristics for approaching proofs and discusses the process students should follow to determine how to proceed from one step to the next through numerous problem solving techniques. A large collection of problems, varying in level of difficulty, are integrated throughout the text and suggested hints for the more challenging problems appear in the instructor's solutions manual and can be used at the instructor's discretion.
  exploring geometry with the geometer's sketchpad answers: Geometry in Action Clark Kimberling, 2003 Geometry in Action uses Sketchpad? to awaken student creativity through discovery-based learning. It supplements any college geometry course in which The Geometer's Sketchpad is used. All students must have access to The Geometer's Sketchpad.Each book is packaged with a CD-ROM for students that illustrates what is meant by geometry in action. Students explore 27 sketches prepared by the author to demonstrate Sketchpad's capabilities by dragging points to see shifts in graphs, by animating tesselations to create new patterns, and much, much more! Also included on this CD is the Poincare Disk, a Sketchpad file used to dig deeper into non-Euclidean geometry with The Geometer's Sketchpad.
  exploring geometry with the geometer's sketchpad answers: MAA Notes , 1983
  exploring geometry with the geometer's sketchpad answers: Roads to Geometry Edward C. Wallace, Stephen F. West, 2015-10-23 Now available from Waveland Press, the Third Edition of Roads to Geometry is appropriate for several kinds of students. Pre-service teachers of geometry are provided with a thorough yet accessible treatment of plane geometry in a historical context. Mathematics majors will find its axiomatic development sufficiently rigorous to provide a foundation for further study in the areas of Euclidean and non-Euclidean geometry. By using the SMSG postulate set as a basis for the development of plane geometry, the authors avoid the pitfalls of many “foundations of geometry” texts that encumber the reader with such a detailed development of preliminary results that many other substantive and elegant results are inaccessible in a one-semester course. At the end of each section is an ample collection of exercises of varying difficulty that provides problems that both extend and clarify results of that section, as well as problems that apply those results. At the end of chapters 3–7, a summary list of the new definitions and theorems of each chapter is included.
  exploring geometry with the geometer's sketchpad answers: A Problem Solving Approach to Mathematics for Elementary School Teachers Rick Billstein, 1997 Sixth ed. also published under title: A problem solving approach to mathematics./ Includes bibliographical references and index.
  exploring geometry with the geometer's sketchpad answers: Teaching Math at a Distance, Grades K-12 Theresa Wills, 2020-10-12 Make Rich Math Instruction Come to Life Online In an age when distance learning has become part of the new normal, educators know that rich remote math teaching involves more than direct instruction, online videos, and endless practice problems on virtual worksheets. Using both personal experience and those of teachers in real K-12 online classrooms, distance learning mathematics veteran Theresa Wills translates all we know about research-based, equitable, rigorous face-to-face mathematics instruction into an online venue. This powerful guide equips math teachers to: Build students’ agency, identity, and strong math communities Promote mathematical thinking, collaboration, and discourse Incorporate rich mathematics tasks and assign meaningful homework and practice Facilitate engaging online math instruction using virtual manipulatives and other concrete learning tools Recognize and address equity and inclusion challenges associated with distance learning Assess mathematics learning from a distance With examples across the grades, links to tutorials and templates, and space to reflect and plan, Teaching Math at a Distance offers the support, clarity, and inspiration needed to guide teachers through teaching math remotely without sacrificing deep learning and academic growth.
  exploring geometry with the geometer's sketchpad answers: Learning and Teaching K-8 Mathematics Janet M. Sharp, Karen Bush Hoiberg, 2005 Real students... Real teaching... Real learning! Try it! This hands-on text takes an ACTIVE approach to teaching mathematics in grades K-8. Activities are embedded in each chapter for adult learners to complete to ensure that they are grasping the mathematics. Completing these activities provides future teachers with invaluable insight into how children think about and learn math, thereby equipping them with the skills to make sound instructional decisions. See it! Throughout the book and on the enclosed Understanding Children's Mathematical Thinking VideoWorkshop CD-ROM, we see coauthor Karen Bush Hoiberg in action. A presidential award-winning elementary teacher, Karen provides a multitude of practical applications and real-world examples to give preservice teachers insight into how children think mathematically. Teach it! The text is organized around four general themes within each major mathematical content area: thinking about the mathematics, learning theories, the role of the teacher, and making algebraic connections. By actually DOING mathematics while analyzing learning theory, novice or preservice teachers are able to connect chapter concepts to actual classroom teaching practices. Many methods books assume that their readers know very little about learning theory and proceed to either overwhelm their readers or trivialize the material through a superficial chapter. Sharp and Hoiberg integrate learning theory throughout their text and show how learning theory can be viewed through mathematics in real classrooms without becoming tedious or distracting. --Chris Ohana, Western Washington University, Field Editor for the NSTA Journal of Science and Children College students planning to be elementary school teachers need a framework and examples that reflect contemporary notions about the teaching and learning of mathematics. Similarly, college methods professors need materials that challenge their students with respect to the mathematical content they are expected to teach and how they can encourage children's mathematical thinking. The needs of both may be met with Learning and Teaching K-8 Mathematics. --Tom Romberg, University of Wisconsin-Madison Perhaps one of the most important aspects of the text is the continual reference to the classroom, seeing it not only in form of vignettes, but hearing it from the voice of a practicing teacher, and also the inclusion of excerpts of questioning techniques used by other practicing teachers. This constant reference to the classroom will help preservice teachers become much more reflective mathematics teachers. --William Fisher, California State University, Chico
  exploring geometry with the geometer's sketchpad answers: A Course in Modern Geometries Judith N. Cederberg, 2013-03-09 A Course in Modern Geometries is designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. Chapter 1 presents several finite geometries in an axiomatic framework. Chapter 2 continues the synthetic approach as it introduces Euclid's geometry and ideas of non-Euclidean geometry. In Chapter 3, a new introduction to symmetry and hands-on explorations of isometries precedes the extensive analytic treatment of isometries, similarities and affinities. A new concluding section explores isometries of space. Chapter 4 presents plane projective geometry both synthetically and analytically. The extensive use of matrix representations of groups of transformations in Chapters 3-4 reinforces ideas from linear algebra and serves as excellent preparation for a course in abstract algebra. The new Chapter 5 uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Each chapter includes a list of suggested resources for applications or related topics in areas such as art and history. The second edition also includes pointers to the web location of author-developed guides for dynamic software explorations of the Poincaré model, isometries, projectivities, conics and fractals. Parallel versions of these explorations are available for Cabri Geometry and Geometer's Sketchpad. Judith N. Cederberg is an associate professor of mathematics at St. Olaf College in Minnesota.
  exploring geometry with the geometer's sketchpad answers: Instructor Manual, Mathematics for Elementary Teachers Via Problem Solving Joanna Osborne Masingila, Frank K. Lester, 2002
  exploring geometry with the geometer's sketchpad answers: Handbook of Research on Instructional Systems and Technology Kidd, Terry T., Song, Holim, 2008-10-31 This book provides information on different styles of instructional design methodologies, tips, and strategies on how to use technology to facilitate active learning and techniques to help faculty and researchers develop online instructional and teaching materials. It enables libraries to provide a foundational reference for researchers, educators, administrators, and others in the context of instructional systems and technology--Provided by publisher.
  exploring geometry with the geometer's sketchpad answers: Children and Computer Technology , 2000
  exploring geometry with the geometer's sketchpad answers: Computer Algebra and Geometric Algebra with Applications Hongbo Li, Peter J. Olver, Gerald Sommer, 2005-06-20 MathematicsMechanization consistsoftheory,softwareandapplicationofc- puterized mathematical activities such as computing, reasoning and discovering. ItsuniquefeaturecanbesuccinctlydescribedasAAA(Algebraization,Algori- mization, Application). The name “Mathematics Mechanization” has its origin in the work of Hao Wang (1960s), one of the pioneers in using computers to do research in mathematics, particularly in automated theorem proving. Since the 1970s, this research direction has been actively pursued and extensively dev- oped by Prof. Wen-tsun Wu and his followers. It di?ers from the closely related disciplines like Computer Mathematics, Symbolic Computation and Automated Reasoning in that its goal is to make algorithmic studies and applications of mathematics the major trend of mathematics development in the information age. The International Workshop on Mathematics Mechanization (IWMM) was initiated by Prof. Wu in 1992, and has ever since been held by the Key L- oratory of Mathematics Mechanization (KLMM) of the Chinese Academy of Sciences. There have been seven workshops of the series up to now. At each workshop, several experts are invited to deliver plenary lectures on cutting-edge methods and algorithms of the selected theme. The workshop is also a forum for people working on related subjects to meet, collaborate and exchange ideas.
  exploring geometry with the geometer's sketchpad answers: Teaching Mathematics Using Popular Culture Elana Reiser, 2015-11-09 Mathematics teachers often struggle to motivate their students. One way to cultivate and maintain student interest is for teachers to incorporate popular media into their methodology. Organized on the subject strands of the Common Core, this book explores math concepts featured in contemporary films and television shows and offers numerous examples high school math teachers can use to design lessons using pop culture references. Outlines for lessons are provided along with background stories and historical references.
  exploring geometry with the geometer's sketchpad answers: Mathematics for Elementary Teachers Via Problem Solving: Instructor's resource manual Joanna Osborne Masingila, 1998
  exploring geometry with the geometer's sketchpad answers: Lines and Curves Victor Gutenmacher, N.B. Vasilyev, 2004-07-23 Broad appeal to undergraduate teachers, students, and engineers; Concise descriptions of properties of basic planar curves from different perspectives; useful handbook for software engineers; A special chapter---Geometry on the Web---will further enhance the usefulness of this book as an informal tutorial resource.; Good mathematical notation, descriptions of properties of lines and curves, and the illustration of geometric concepts facilitate the design of computer graphics tools and computer animation.; Video game designers, for example, will find a clear discussion and illustration of hard-to-understand trajectory design concepts.; Good supplementary text for geometry courses at the undergraduate and advanced high school levels
  exploring geometry with the geometer's sketchpad answers: Mathematics Catalog 2005 Neil Thomson, 2004-10
  exploring geometry with the geometer's sketchpad answers: Navigating Through Geometry in Grades 3-5 M. Katherine Gavin, 2001 Navigating through geometry in grades 3-5.
  exploring geometry with the geometer's sketchpad answers: New Horizons in Mathematics and Science Education , 2001
  exploring geometry with the geometer's sketchpad answers: The Foundations of Geometry Gerard Venema, 2006 For sophomore/junior-level courses in Geometry; especially appropriate for students that will go on to teach high-school mathematics. This text comfortably serves as a bridge between lower-level mathematics courses (calculus and linear algebra) and upper-level courses (real analysis and abstract algebra). It fully implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers. Foundations of Geometry particularly teaches good proof-writing skills, emphasizes the historical development of geometry, and addresses certain issues concerning the place of geometry in human culture.
  exploring geometry with the geometer's sketchpad answers: Teaching Mathematics for the 21st Century Linda Huetinck, Sara N. Munshin, 2008 This third edition of T eaching Mathematics for the 21st Century continues to help teachers let the secret out-to open up to their students the wonderful discoveries and challenges of the pattern-making and problem-solving aspects of a fascinating subject: mathematics. The rationale remains the same-to enable prospective and current teachers to access and use tools and strategies to effectively teach mathematics to contemporary students. Changing demographics, knowledge of how people learn, and technology all impact the way we educate our young people. This edition incorporates lessons and strategies from programs that have proven success in many types of classrooms. Many of these examples help students connect mathematics to real life situations and communicate their understanding of the underlying concepts. Although technology is constantly being upgraded, ways to increase student motivation through its application remains a goal. For example--since applets can enhance a lesson whether the teacher uses a computer projector, a smart board, or has students work individually on computers--we have identified several sources of mathematics applets that can be correlated to various lessons. Research citations and summaries have been updated to reflect current information on teaching and learning. For future teachers.
  exploring geometry with the geometer's sketchpad answers: Discovering Geometry Michael Serra, Key Curriculum Press Staff, 2003-03-01
  exploring geometry with the geometer's sketchpad answers: Teaching and Learning Mathematics Linda Jensen Sheffield, Douglas E. Cruikshank, 2004-03-29 Help students make sense of mathematics Rather than merely discussing how to improve students’ ability to do mathematics, this fifth edition focuses on helping them make sense of mathematics. Based on research on the functioning of the mind as it engages in learning, the text supports teachers as they promote mathematical understanding, strengthen students’ abilities to think, and help students to attain computational fluency. Features A rich collection of ready-to-use learning activities Fully integrated language and intent of Principles and Standards for School Mathematics (PSSM). A greater emphasis on problem solving and higher-level thinking A greater focus on teaching mathematics to diverse learners Descriptions of a variety of promising and effective mathematics programs for the K – 8 levels
  exploring geometry with the geometer's sketchpad answers: HCI International 2013 - Posters' Extended Abstracts Constantine Stephanidis, 2013-06-13 This is the first of a two-volume set (CCIS 373 and CCIS 374) that constitutes the extended abstracts of the posters presented during the 15th International Conference on Human-Computer Interaction, HCII 2013, held in Las Vegas, USA, in July 2013, jointly with 12 other thematically similar conferences. The total of 1666 papers and 303 posters presented at the HCII 2013 conferences was carefully reviewed and selected from 5210 submissions. These papers address the latest research and development efforts and highlight the human aspects of design and use of computing systems. The papers accepted for presentation thoroughly cover the entire field of human-computer interaction, addressing major advances in knowledge and effective use of computers in a variety of application areas. The extended abstracts were carefully reviewed and selected for inclusion in this two-volume set. The papers included in this volume are organized in the following topical sections: HCI design approaches, methods and techniques; usability methods, techniques and studies; universal access and eInclusion; multimodal and ambient interaction; cognitive and psychological aspects of interaction; perception and interaction; ergonomic and human modelling issues; capturing gaze, biosignals and brainwaves; development environments; product design, marketing and advertisement.
  exploring geometry with the geometer's sketchpad answers: Exploring Geometry with The Geometer's Sketchpad Dan Bennett, 2002-01-01 Blackline activity masters for use with The Geometer's Sketchpad--Cover.
  exploring geometry with the geometer's sketchpad answers: How Students Think When Doing Algebra Steve Rhine, Rachel Harrington, Colin Starr, 2018-11-01 Algebra is the gateway to college and careers, yet it functions as the eye of the needle because of low pass rates for the middle school/high school course and students’ struggles to understand. We have forty years of research that discusses the ways students think and their cognitive challenges as they engage with algebra. This book is a response to the National Council of Teachers of Mathematics’ (NCTM) call to better link research and practice by capturing what we have learned about students’ algebraic thinking in a way that is usable by teachers as they prepare lessons or reflect on their experiences in the classroom. Through a Fund for the Improvement of Post-Secondary Education (FIPSE) grant, 17 teachers and mathematics educators read through the past 40 years of research on students’ algebraic thinking to capture what might be useful information for teachers to know—over 1000 articles altogether. The resulting five domains addressed in the book (Variables & Expressions, Algebraic Relations, Analysis of Change, Patterns & Functions, and Modeling & Word Problems) are closely tied to CCSS topics. Over time, veteran math teachers develop extensive knowledge of how students engage with algebraic concepts—their misconceptions, ways of thinking, and when and how they are challenged to understand—and use that knowledge to anticipate students’ struggles with particular lessons and plan accordingly. Veteran teachers learn to evaluate whether an incorrect response is a simple error or the symptom of a faulty or naïve understanding of a concept. Novice teachers, on the other hand, lack the experience to anticipate important moments in the learning of their students. They often struggle to make sense of what students say in the classroom and determine whether the response is useful or can further discussion (Leatham, Stockero, Peterson, & Van Zoest 2011; Peterson & Leatham, 2009). The purpose of this book is to accelerate early career teachers’ “experience” with how students think when doing algebra in middle or high school as well as to supplement veteran teachers’ knowledge of content and students. The research that this book is based upon can provide teachers with insight into the nature of a student’s struggles with particular algebraic ideas—to help teachers identify patterns that imply underlying thinking. Our book, How Students Think When Doing Algebra, is not intended to be a “how to” book for teachers. Instead, it is intended to orient new teachers to the ways students think and be a book that teachers at all points in their career continually pull of the shelf when they wonder, “how might my students struggle with this algebraic concept I am about to teach?” The primary audience for this book is early career mathematics teachers who don’t have extensive experience working with students engaged in mathematics. However, the book can also be useful to veteran teachers to supplement their knowledge and is an ideal resource for mathematics educators who are preparing preservice teachers.
  exploring geometry with the geometer's sketchpad answers: ENC Focus , 2000
  exploring geometry with the geometer's sketchpad answers: APOS Theory Ilana Arnon, Jim Cottrill, Ed Dubinsky, Asuman Oktaç, Solange Roa Fuentes, Maria Trigueros, Kirk Weller, 2013-08-04 In spite of the fact that APOS Theory has been used extensively in numerous scholarly publications, in the design of textbooks, and in teaching practice, there is no single references that contains all the relevant information about its components, and provides guidance about its application. The goal of this book is to present the main elements of APOS theory. It should be useful for researchers who work with, or would like to learn more about, this theoretical approach, people who are interested in the way which mathematical conceptions are constructed according to this theory, Mathematics Education researchers, graduate students in Mathematics Education, and Mathematics instructors.
  exploring geometry with the geometer's sketchpad answers: Algebra 2 McDougal Littell Incorporated, Ron Larson, 2004
  exploring geometry with the geometer's sketchpad answers: Mathematics and Science for Students with Special Needs Eisenhower National Clearinghouse for Mathematics and Science Education, 2003
  exploring geometry with the geometer's sketchpad answers: THE Journal , 1992
  exploring geometry with the geometer's sketchpad answers: Proceedings of the Fifth Annual International Conference on Technology in Collegiate Mathematics Lewis Lum, 1994 Over 1500 mathematicians from around the world attended The Fifth Annual International Confererence on Technology in Collegiate Mathematics. The conference theme was Exploring the Mathematics Learning Environment of Tomorrow. Among the many conference highlights were lectures on: applying systems engineering to education; how contents of courses have changed in this computer era; and real mathematics, rational computers, and complex people.
Exploring - Discover Your Future
Exploring helps teens and young adults discover career opportunities and life goals within a group of friends and real-world, one-on-one mentorship.

EXPLORING Definition & Meaning - Merriam-Webster
The meaning of EXPLORE is to investigate, study, or analyze : look into —sometimes used with indirect questions. How to use explore in a sentence.

EXPLORING | English meaning - Cambridge Dictionary
EXPLORING definition: 1. present participle of explore 2. to search a place and discover things about it: 3. to think…. Learn more.

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Exploring - definition of exploring by The Free Dictionary
To investigate systematically; examine: explore every possibility. 2. To search into or travel in for the purpose of discovery: exploring outer space. 3. Medicine To examine (a body cavity or …

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Find 13 different ways to say EXPLORING, along with antonyms, related words, and example sentences at Thesaurus.com.

EXPLORING definition in American English | Collins English ...
EXPLORING definition: to examine or investigate , esp systematically | Meaning, pronunciation, translations and examples in American English

Exploring - Discover Your Future
Exploring helps teens and young adults discover career opportunities and life goals within a group of friends and …

EXPLORING Definition & Meaning - Merriam-Webster
The meaning of EXPLORE is to investigate, study, or analyze : look into —sometimes used with indirect …

EXPLORING | English meaning - Cambridge Dictionary
EXPLORING definition: 1. present participle of explore 2. to search a place and discover things about it: 3. to …

Exploring
Exploring is the parent company of several unique, trailblazing Atlanta-based companies. Our companies …

Exploring - definition of exploring by The Free Diction…
To investigate systematically; examine: explore every possibility. 2. To search into or travel in for the purpose of …