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    Dynamic Geometry Game for Pods
    Dynamic Geometry Game for Pods
    Dynamic Geometry Game for Pods
    Ebook264 pages1 hour

    Dynamic Geometry Game for Pods

    By Gerry Stahl

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    About this ebook

    This book contains adventures in digital geometry for the minds of students in pods and in home-schooling. Learning about geometry has inspired many of the most important thinkers for centuries and helped them to make sense of the world. This sequence of 50 hands-on challenges will step learners through the most exciting experiences of geometry, from basic points, lines and circles to construction and proof. The book is structured as a game: a series of thought-provoking challenges that provides a stimulating experience of collaboration with pod-mates and a fun introduction to geometry.
    LanguageEnglish
    PublisherLulu.com
    Release dateAug 21, 2020
    ISBN9781716638282
    Dynamic Geometry Game for Pods
    Author

    Gerry Stahl

    Gerry Stahl's professional research is in the theory and analysis of CSCL (Computer-Supported Collaborative Learning). In 2006 Stahl published "Group Cognition: Computer Support for Building Collaborative Knowledge" (MIT Press) and launched the "International Journal of Computer-Supported Collaborative Learning". In 2009 he published "Studying Virtual Math Teams" (Springer), in 2013 "Translating Euclid," in 2015 a longitudinal study of math cognitive development in "Constructing Dynamic Triangles Together" (Cambridge U.), and in 2021 "Theoretical Investigations: Philosophical Foundations of Group Cognition" (Springer). All his work outside of these academic books is published for free in volumes of essays at Smashwords (or at Lulu as paperbacks at minimal printing cost). Gerry Stahl earned his BS in math and science at MIT. He earned a PhD in continental philosophy and social theory at Northwestern University, conducting his research at the Universities of Heidelberg and Frankfurt. He later earned a PhD in computer science at the University of Colorado at Boulder. He is now Professor Emeritus at the College of Computation and Informatics at Drexel University in Philadelphia. His website--containing all his publications, materials on CSCL and further information about his work--is at http://GerryStahl.net.

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      Book preview

      Dynamic Geometry Game for Pods - Gerry Stahl

      Welcome

      These days, much student learning takes place in small pods of students working together. Often, they interact and communicate online. In addition, students engage in home-schooling, drawing upon online resources and media.

      Online and pod-based education opens new opportunities for highly motivating and effective approaches. However, success requires innovative and well-designed curriculum. The present Dynamic Geometry Game for Pods translates the learning of traditional Euclidean geometry into an engaging, stimulating and collaborative experience for online pods of students or for individual home-schooled students.

      Dynamic geometry is a recent transformation of classic geometry into an online app, which allows one to explore geometric figures by dragging them around the computer screen. Students can construct their own figures and receive immediate automated feedback about the results. This can provide a lively, hands-on experience of geometry.

      A free computer app, GeoGebra, is available at: www.geogebra.com. GeoGebra now includes a Class mode that is ideal for small pods of students working together under a teacher’s supervision. GeoGebra student apps and teacher Class dashboard can be shared in a Zoom session if desired. The Dynamic Geometry Pod Game can be opened at: https://www.geogebra.org/m/vhuepxvq#material/swj6vqbp. The game can be played immediately then.

      If you would like to print out a copy of the game – perhaps to take notes in – this pdf version is available at: http://gerrystahl.net/elibrary/game/game.pdf.

      Since the beginning of Western civilization 2,500 years ago, geometry has trained students in rigorous thinking. Perhaps dynamic geometry can help the next generation enhance their understanding of today’s complex world.

      At the end of this volume is an academic article that discusses how this game can be a model of curriculum for blended learning, which combines teacher-led classroom instruction and student-centered collaborative learning. It was published as: Stahl, G. (2021). Redesigning mathematical curriculum for blended learning. Education Sciences. 11(165), pages 1-12. Web: https://www.mdpi.com/2227-7102/11/4/165.

      Contents

      Welcome      5

      Contents      6

      Intro for Adventurous Students      9

      Intro for Parents and Teachers      11

      Game Part A      13

      LEVEL 1. BEGINNER LEVEL      13

      Challenge 1: Play House      14

      Challenge 2: Dynamic Stick Figures      15

      Challenge 3: Play around with Points, Lines and Circles      16

      LEVEL 2: CONSTRUCTION LEVEL      17

      Challenge 4: Play by Dragging Connections      17

      Challenge 5: Play with Hidden Objects      18

      Challenge 6. Construct Polygons in Different Ways      19

      LEVEL 3: TRIANGLE LEVEL      20

      Challenge 7: Construct an Equilateral Triangle      20

      Challenge 8: Find Dynamic Triangles      21

      LEVEL 4: CIRCLE LEVEL      22

      Challenge 9: Construct the Midpoint      22

      Challenge 10: Construct a Perpendicular Line      23

      Challenge 11: Construct a Parallel Line      24

      Game Part B      26

      LEVEL 5: DEPENDENCY LEVEL      26

      Challenge 12: Triangles with Dependencies      27

      Challenge 13: An Isosceles Triangle      28

      Challenge 14: A Right Triangle      29

      Challenge 15: An Isosceles-Right Triangle      30

      LEVEL 6. COMPASS LEVEL      30

      Challenge 16: Copy a Length      31

      Challenge 17: Use the Compass Tool      32

      Challenge 18: Make Dependent Segments      33

      Challenge 19: Add Segment Lengths      34

      Challenge 20: Copy vs. Construct a Congruent Triangle      35

      Challenge 21: Construct a Congruent Angle      36

      Game Part C      38

      LEVEL 7: CONGRUENCE LEVEL      38

      Challenge 22: Combinations of Sides and Angles of Triangles      39

      Challenge 23: Side-Side-Side (SSS)      40

      Challenge 24: Side-Angle-Side (SAS)      41

      Challenge 25: Angle-Side-Angle (ASA)      42

      Challenge 26: Side-Side-Angle (SSA)      43

      LEVEL 8. INSCRIBED POLYGON LEVEL      44

      Challenge 27: The Inscribed Triangles Challenge Problem      44

      Challenge 28: Inscribed Squares      45

      Challenge 29: Prove Inscribed Triangles      46

      Game Part D      48

      LEVEL 9: TRANFORMATION LEVEL      48

      Challenge 30: Translate by a Vector      49

      Challenge 31: Reflect About a Line      50

      Challenge 32: Rotate Around a Point      51

      Challenge 33: Combine Transformations      52

      Challenge 34: Create Dynamic Patterns      53

      LEVEL 10. QUADRILATERAL LEVEL      53

      Challenge 35: Construct Quadrilaterals with Constraints      54

      Challenge 36: Construct a Rhombus      55

      Challenge 37: Quadrilateral Areas      56

      Challenge 38: Build a Hierarchy of Quadrilaterals      57

      Game Part E      59

      LEVEL 11: ADVANCED GEOMETER LEVEL      59

      Challenge 39: The Centroid of a Triangle      60

      Challenge 40: The Circumcenter of a Triangle      61

      Challenge 41: The Orthocenter of a Triangle      62

      Challenge 42: The Incenter of a Triangle      63

      Challenge 43: The Euler Segment of a Triangle      64

      Challenge 44: The Nine-Point Circle of a Triangle      65

      LEVEL 12: PROBLEM SOLVER LEVEL      66

      Challenge 45: Treasure Hunt      66

      Challenge 46: Square and Circle      67

      Challenge 47: Cross an Angle      68

      LEVEL 13: EXPERT LEVEL      68

      Challenge 48: How Many Ways Can You Invent?      69

      Challenge 49: Dependencies in the World      70

      Challenge 50: Into the Future      71

      Extra Bonus Dynamic Geometry      72

      A Special Challenge      72

      Visualizing the World’s Oldest Theorem      73

      Proof of Special Challenge      83

      Proof Involving the Incenter of a Triangle      83

      Your own Custom Geometry      90

      Transforming a Factory      91

      Navigating Taxicab Geometry      96

      Redesigning Mathematical Curriculum for Blended Learning      99

      Introduction: Student Pods during the Pandemic      99

      Designing for Virtual Math Teams      101

      Redesigning for Pandemic Pods with GeoGebra Classes      103

      Findings from VMT Trials      107

      Supporting Group Practices in Blended Learning      108

      Broadening the Model for Blended Learning      113

      References      115

      Notes      119

      Intro for Adventurous Students

      The Dynamic Geometry Game for Pods is a series of Challenges for your pod to construct interesting and fun geometric figures. Many of the figures will have hidden features and your pod will learn how to design them. So put together your Pod with three, four, five or six people from anywhere in the world who want to play the game together online.

      The Game consists of several levels of play, each with a set of Challenges to do together online. The Challenges in the beginning levels do not require any previous knowledge about geometry or skill in working together. Playing the Challenges in the order they are given will prepare you with everything you need to know for the more advanced levels. Be creative and have fun. See if you can invent new ways to do the Challenges.

      Each Challenge has questions to think about and answer. These will help you to make sense of the Challenges and your solutions. Your responses to the questions will help your teammates in your pod to understand what you discovered about the Challenge and to know what you would like help understanding. Be sure to answer the questions and to read the answers from the rest of your pod. Try each Challenge at your level until everyone in your pod understands how to meet the Challenges. Then move on to the next level. Take your time until everyone has mastered the level. Then agree as a team to go to the next level. Most levels assume that everyone has mastered the previous level. The levels become harder and harder – see how far your pod can go.

      Geometry has always been about constructing dependencies into geometric figures and discovering relationships that are therefore necessarily true and provable. Dynamic geometry (like GeoGebra) makes the construction of dependencies clear. The game Challenges at each level will help you to think about geometry this way and to design constructions with the necessary dependencies. The sequence of levels is designed to give you the knowledge and skills you need to think

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