Geometric Structures
An Inquiry-Based Approach for Prospective Elementary and Middle School Teachers
Häftad, Engelska, 2007
2 279 kr
Produktinformation
- Utgivningsdatum2007-04-25
- Mått100 x 100 x 100 mm
- Vikt100 g
- SpråkEngelska
- Antal sidor688
- Upplaga1
- FörlagPearson Education
- EAN9780131483927
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John Wolfe, Oklahoma State University: John recently was promoted to professor emeritus at Oklahoma State University. After graduating from the University of California at Berkeley in 1971 he was active in mathematics research including several publications and National Science Foundation research grants in Banach Space Theory. His enthusiasm for teaching has been recognized by awards from both the Mathematical Association of America and the Regents of Oklahoma State University. Over the past several years educational issues with a special passion for geometry have been the focus of his professional life. Woodworking, travel, camping and grandkids are becoming increasingly important in his life. Douglas B. Aichele, Oklahoma State University: Douglas grew up in Great Neck, NY. He received his undergraduate and graduate education in mathematics at the University of Missouri-Columbia. He has been a faculty member at Oklahoma State University for many years and was appointed Regents Professor in 1989. He is currently serving as Associate Head of the Department of Mathematics. Good teaching of mathematics has always been important to him and he has been recognized over the years by such institutions/organizations as the OSU Blue Key Alumni Association, Mathematical Association of America, and the University of Missouri-Columbia.His other interests reside mainly in the outdoors - particularly spending time with his family in the mountains of Colorado at a log cabin that he built by hand. He is an avid backpacker and hiker; he has climbed most of Colorado's Fourteeners (mountains with elevations greater then 14,000 ft.) and several mountains and glaciers near Juneau, Alaska.
- Part I: Paper FoldingChapter 0 - Warm Up Activities0.0 Introduction0.1 Folding Polygons from a Circle0.2 Making Squares0.3 Two Congruent Halves0.4 Dissecting FiguresChapter 1 - Polygons and the Angle Relationships1.0 Introduction1.1 Parallel Line Grid - Triangle Angle Sum1.2 Envelope Fold - Triangle Angle Sum1.3 Triangle and Quadrilateral Angle Sums by Tearing1.4 Polygon Angle Sums: How many Triangles?1.5 The Angles of a Polygon1.6 When Does Erika's Idea Work?1.7 The Greedy Triangle1.8 Problems: Angle Sums and Angle Relationships1.9 Four Kinds of Related Angles1.10 Figuring Angles and Checking by Measurement1.11 Parallel Lines: How to Recognize Them1.12 Measuring Sides and Angles of Triangles1.14 Convex: Different Ways to Make Sense of It1.14a Angle Problems - Version A1.14b Angle Problems - Version B1.15 Angle Probems - More1.16 How Do I Know if I Understand?1.17 Conjecturing ABout Quadrilaterals1.18 Possible or Not?1.19 True or False (with Example)1.20 Under What Conditions? Chapter 2 - Quadrilaterals and Their Definitions2.0 Introduction2.1 Checking Properties of Quadrilaterals2.2 Properties of Quadrilaterals2.3 Marking Quadrilateral Properties2.4 Properties of Diagonals of Quadrilaterals2.5 Checking Quadrilaterals by Folding2.6 Read Carefully: Every Word Counts!2.7 Checking Examples Visually or Physically2.8 Exploring Medial Quadrilaterals2.9a Problems: Properties of Quadrilaterals, Version A2.9b Problems: Properties of Quadrilaterals, Version B2.10 More Problems: Properties of Quadrilaterals2.11 A Deeper Understanding of Definitions2.12 Special Cases of Quadrilaterals2.13 Definitions: Inclusive or Exclusive2.14 Problems: Inclusive and Exclusive Definitions2.15 What Is a Kite? Equivalent Definitions2.16a Problems: Definitions of Quadrilaterals, Version A2.16b Problems: Definitions of Quadrilaterals, Version B2.17 More Problems: Definitions of Quadrilaterals2.18 How Do I Know if I Understand? Prologue:Four Contexts for Geometric ConstructionsPrologue to Chapters 3, 10, 12, and 14 Chapter 3 - Constructions by Paper Folding3.0 Introduction3.1 Introducing CDs: Two Basic Constructions3.2 CD Problem: A Parallel Line3.3 CD Problem: The Median3.4 CD Problem: An Equilateral Triangle3.5 CD Problem: A Square3.6 Circumscribing Circle3.7 Inscribed Circle3.8 Balance Point of a Triangle3.9 Additional CD Problems Using Basic Construction Steps3.10 Group Problem: Inscribed Circles3.11 Folding a Six-Pointed Star or a "Snowflake"3.12 Problems Involving Paper Folding3.13 How Do I Know if I Understand? Chapter 4 - Explorations in Three-dimensional Geometry4.0 Introduction4.1 Polyhedra (Solids) from an Envelope4.2 Roll-and-Fold Prism and Pyramid Activities4.3 Net Project A: Prisms4.4 Prisms4.5 Makiing Sense of Volume: A Basic Relationship4.6 Net Project B: Pyramids4.7 Pyramids4.8 Edges, Faces, and Vertices of Polyhedra4.9 Special Kinds of Polyhedra4.10 Riddles with Solids4.11 Volumes Prisms, Pyramids, and Spheres4.12 Volume of a Pyramid4.13 What Does Volume Really Mean?4.14 Volume of Solids: First Try4.15a Solid-Geometry Problems, Version A4.15b Solid-Geometry Problems, Version B4.16 More Solid-Geometry Problems Addendum:Unit Origami: An Introduction4.17 Instructions for the Basic Parallelogram Unit4.18 Project for the Whole Class: Monster Stellated Icosahedron4.19 Unit Origami Projects4.20 Some Geometry of Unit Origami4.21 Convex Deltahedra: How Many Are There?4.22 Problems: Unit Origami4.23 How Do I Know if I Understand? Part2 GeoBoards and Dot Paper Chapter 5 - Area5.0 Introduction5.1 How Much Space in a Triangle?5.2 Areas on a Geoboard5.3 Two Ways: Cut-up and Take-away5.4 Areas: Parallelograms and Trapezoids5.5 Area by Julie's Way5.6 Which Ways Work for These Figures?5.7 Areas: How Many Ways?5.8 Area Problems: First Try5.9 A Sampling of Area Problems5.10 Making Sense of Common Units for Length and Area5.11a Area Problems, Version A5.11b Area Problems, Version B5.12 More Area Problems5.13 How Do I Know if I Understand? Chapter 6 - Explorations with Geoboard Areas6.0 Introduction6.1 Areas of Skew Quadrilaterals6.2 Solid Tile Shapes6.3 Problems: Tile Shapes6.4 Areas of Tile Shapes6.5 Areas by Counting Pets6.6 How Many Tile Shapes with Five Squares?6.7 Counting Areas: Pick's Formula6.8 Skew Figures6.9 Discovering, Describing, and Using Relationships6.10 Sean's Idea: Area = Inside Pegs6.11a Problems: Geoboard Areas, Version A6.11b Problems: Geoboard Areas, Version B6.12 More Problems: Geoboard Areas6.13 How Do I Know if I Understand? Chapter 7 - Similarity and Slope7.0 Introduction7.1 Slope or Steepness7.2 Slope: Parallel and Perpendicular7.3 Slope Problems, Part 17.4 Slope Problems, Part 27.5 Linear Equations, Tables of Values, and Slopes7.6 Similar Figures and Their Properties7.7 Similar Figures and Proportionality7.8 Measuring Proportionality7.9 Reasoning withSimilar Triangles7.10 Similarity and Scale Factors (Length Factors)7.11 Scaling, Areas, and Area Factors7.12 Scaling Problems, First Try7.13 Scaling Problems7.14 Scaling and Volume of Solids7.15a Problems: Slope, Similarity, and Scaling, Version A7.15b Problems: Slope, Similarity, and Scaling, Version B7.16 More Problems onSlope, Similarity, and Scaling7.17 How Do I Know if I Understand? Chapter 8 - Pythagorean Theorem and Perimeter8.0 Introduction8.1 RightTriangles of Squares8.2 Pythagorean Puzzles8.3 Estimating Perimeters on a Geoboard8.4 Slant Lengths on a Geoboard8.5 Geoboard Perimeters8.6 Three Special Triangles8.7 Pythagorean Problems, First Try8.8a Perimeter and Right-Triangle Problems, Version A8.8b Perimeter and Right-Triangle Problems, Version B8.9 More Perimeter and Right-Triangle Problems8.10 How Do I Know if I Understand? Chapter 9 - Geometry of Circles9.0 Introduction9.1 Perimeter (Circumference) of a Circle9.2 Area of a Circle9.3 Area and Perimeter of Circles and Sectors9.4 Area Problems with Circles, First Try9.5 Problems: Area and Perimeter of Circles9.6 Inscribed Angles of Arcs of Circles9.7 The Law of Thales9.8 Circumscribed or Cyclic Polygons9.9 Circumscribing Circle for a Cyclic Quadrilateral9.10 Problems: Inscribed Angles and Circumscribed Polygons9.11a Problems: Geometry of Circles Version A9.11b Problems: Geometry of Circles, Version B9.12 Revisiting Volumes: Cones and Cylinders9.13 Surface Area of an Orange9.14 More Problems: Geometry of Circles9.15 How Do I Know if I Understand? Part 3 - Straightedge and Compass Chapter 10 - Straightedge and Compass Constructions10.0 Introduction10.1 Basic Straightedge and Compass Constructions10.2 Straightedge and Compass: Construct a Parallel Line10.3 Examples: Reasoning in Construction Problems10.4 Reasoning in Construction Problems10.5 Making Triangles, I:Side-Side-Side10.6 Making Triangles, II: Side-Angle-Side10.7 Making Triangles, III: Angle-Side-Angle10.8 Making Triangles, IV: Side-Side-Angle (Ambiguous Case)10.9 Congruence Conditions for Triangles10.10 How Do I Know I Understand? Chapter 11 - Congruence Conditions and Reasoning from Definitions to Properties11.0 Introduction11.1 Congruence Conditions for Triangles and CPCT11.2 Problems:Congruence Conditions and CPCT11.3 Justifications by Congruence Conditions11.4a Problems: Congruence Conditions, Version A11.4b Problems: Congruence Conditions, Version B11.5 More Problems: Congruence Conditions11.6 FromDefinitions to Properties: Five-Step Reasoning11.7 Example: Five-Step Reasoning, Problem A11.8 Five Step reasoning, First Try11.9 More Problems Using Five-Step Reasoning11.10 How Do I Know if I Understand? Part 4 - Computer Constructions and Explorations Chapter 12 - Computer Constructions12.0 Introduction12.1 Getting Started with Computer Construction Software12.2 Constructing Objects: Midpoints12.3 Constructing Objects: Bisectors12.4 Constructing Objects: Altitudes and Medians12.5 The Euler Line of a Triangle12.6 The Nine-point Circle of a Triangle12.7 The Medial Quadrilateral of Quadrilateral12.8 Problems: Investigating Relationships by Using Geometric Properties12.9 How Do I Know if I Understand? Chapter 13 - Computer Explorations13.0 Introduction13.1 Triangle Inequalities13.2 Angle Bisectors: Why the Incenter Works13.3 Perpendicular Bisectors: Why the Circumcenter Works13.4 Medians and the Centroid of a Triangle13.5 Altitudes: The Orthic Triangle13.6 Angle Bisectors, Medians, and Altitudes: Some Relationships13.7 Revisiting the Medial Triangle: Perimeter and Area13.8 Revisiting the Medial Quadrilateral: Area13.9 Quadrilaterals and Circles13.10 Circles: Central Angles and Inscribed Angles13.11 Circles: More on Inscribed Angles and Arcs13.12 Problems: Investigating Relationships by Using Number Ideas13.13 How Do I Know if I Understand? Part 5 - Mira (Reflecta) and Tracing Paper Chapter 14 - Mira Contructions14.0 Introduction14.1 The Mira: What Does it Mean?14.2 Reflection Lines and Point-Image Segments14.3 Constructions with a Mira (CDs)14.4 Altitudes of a Triangle14.5 Altitudes, Orthocenters, and Trapezoids14.6 Altitude Constructions with a Mira14.7 Measuring a Triangle's Three Altitudes14.8 Where is the Circumcenter?14.9 How Do I Know if I Understand? Chapter 15 - Symmetry15.0 Introduction15.1 Miniproject: Fold-andCut Paper Figures15.2 Fold-and-Cut (Symmetric) Shapes15.3 Orientation: One or Two Sides?15.4a Problems: Symmetry, Version A15.4b Problems: Symmetry, Version B15.5 Fold and Cut: Three Symmetry Lines15.6 Fold and Cut: Fivefold Symmetry15.7 Problems: More on Symmetry15.8 How Do I Know if I Understand? Chapter 16 - The Four Symmetries16.0 Introduction16.1 Four Actions: Slide, Flip, Turn, and Glide-Flip16.2 Four Symmetries16.3 Translations and Coordinates16.4 Problems: Four Actions or Symmetries16.5 Combinatons of Reflections16.6 Actions: Which of the Four Types?16.7 Rotations and Glide-Reflections: Point-Image Segments16.8 How Do You Get from One to the Other?16.9 CD Problem: Find the Center of Rotation16.10 CD Problem: Find the Glide-Reflection Line16.11 An Experiment with the Four-Kinds Principle16.12 Marking Symmetries on Wallpaper Designs16.13a Problems: Four Types of Symmetry, Version A16.13b Problems:Four Types of Symmetry, Version B16.14 More Problems Involving the Four Types of Symmetry16.15 How Do I Know if I Understand? Prologue: Symmetries of Decorative ArtPrologue to Chaptes 17, 18, and 19 Chapter 17 - Symmetries of Mandalas17.0 Introduction17.1 Symmetries of Mandalas17.2 Classifying Mandalas, First Try17.3 Classifying Mandalas17.4 Mandalas: One or Two Sides?17.5 Template Design Mandalas17.6 Template Design Problems17.7 Express Yourself with a Mandala17.8 The Symmetry Classification of Mandalas17.9 Problems: Mandalas17.10a Problems: Mandalas, Version A17.10b Problems: Mandalas, Version B17.11 How Do I Know if I Understand? Chapter 18 - Symmetries of Borders18.0 Introduction18.1 Glide-Reflectional and Half-turn Symmetry18.2 Classifying Borders, First Try18.3 Borders: What Is Their Symmetry Type?18.4 Generating Borders18.5 Borders: Make Your Own Display18.6 The Symmetry Classificaton of Borders18.7 Problems Classifying Borders18.8a Problems: Borders, Version A18.8b Problems: Borders, Version B18.9 How Do I Know if I Understand? Chapter 19 - Escher-Style Tessellations19.0 Introduction19.1 Escher Tessellations, Type TTTT19.2 How to Make a Type TTTT Tessellation19.3 Cut and Tape: Make Your Own Tessellating Shape19.4 Miniproject: Recognizability19.5 Four Moves for Tessellating Squares19.6 What Are the Possible Heesch Types?19.7 What is the Heesch Type?19.8 Project: Making Escher-Style Tessellations19.9 Checking Understanding of Heesch Types19.10 Marking Symmetries on Escher Tessellations19.11 Do These Tessellations Work?19.12 How Do I Know if I Understand? AppendicesA.1 A Guide for You, the Student: Making Sense of Geometry in an Inquiry-based ClassA.2 GeoSET Website: Internet Resources for StudentsA.3 Construct/Describe ProblemsA3.1 Hints for Doing CD ProblemsA3.2 Shorthand Comments for CD ProblemsA3.3 Catalogue of CD ProblemsA.4 Dot Paper Template for Copying Bibliography Index