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students, or groups of students, as independent projects, which can be combined in a final class exhibit. Many students are fascinated with M. C. Escher’s graphic work, which involves tiling with figures such as lizards, birds, or angels.A report on Escher could be one of the independent projects. Note that tilings provide interesting infinite figures for students to analyze from the point of view of symmetry.To support and encourage that inquiry, the same symmetry questions are asked about each tiling in each discussion section.That kind of analysis can be a worthwhile complement to the projects suggested above. Although tiling is as ancient as tiles, it is a fairly recent topic in mathematics. Except for Kepler, all of the research on it has occurred since the nineteenth century, and new results continue to appear today. The definitive reference on the mathematics of tiling is Tilings and Patterns by Banko Grunbaum and G. C. Shephard (Freeman, N.Y., 1987), which features many beautiful figures. See page 203 for teacher notes to this section. 98 Section 7 Tiling Geometry Labs © 1999 Henri Picciotto, www.MathEducationPage.org
LAB 7.1 Tiling with Polyominoes Name(s) Equipment: Grid paper Imagine a flat surface, extending forever in all directions. Such a surface is called a plane. If you had an unlimited supply of rectangular polyomino tiles, you could cover as large an area as you wanted. For example, you could lay out your tiles like this: . . . or like this: 1. Find interesting ways to tile a plane with rectangular polyominoes. Record them on your grid paper. Below are two bent tromino tilings: The first one is made of two-tromino rectangles; the second is made of diagonal strips. By extending the patterns, you can tile a plane in all directions. 2. Can you extend this U pentomino tiling in all directions Experiment and explain what you discover. 3. Find ways to tile a plane with each of the following tetrominoes. a. n b. l c. t 4. Find ways to tile a plane with each of the twelve pentominoes. Geometry Labs Section 7 Tiling 99 © 1999 Henri Picciotto, www.MathEducationPage.org
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GEOMETRY Henri Picciotto LABS ACTIV
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Project Editor: Editorial Assistant
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Contents Introduction .............
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9 Distance and Square Root ........
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Introduction About This Book This b
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• Manipulatives can motivate stud
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If you have access to computers, I
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1 Angles An essential foundation of
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LAB 1.1 Angles Around a Point Name(
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LAB 1.2 Angle Measurement (continue
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LAB 1.4 Angles of Pattern Block Pol
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LAB 1.5 Angles in a Triangle Name(s
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LAB 1.6 The Exterior Angle Theorem
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LAB 1.6 The Exterior Angle Theorem
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LAB 1.7 Angles and Triangles in a C
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LAB 1.8 The Intercepted Arc (contin
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LAB 1.10 Soccer Angles Name(s) Equi
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LAB 1.10 Soccer Angles (continued)
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LAB 1.10 Soccer Angles (continued)
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2 Tangrams Tangram puzzles are quit
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LAB 2.1 Meet the Tangrams (continue
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LAB 2.2 Tangram Measurements (conti
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LAB 2.4 Symmetric Polygons Name(s)
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3 Polygons In the first two labs, t
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LAB 3.1 Triangles from Sides (conti
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LAB 3.2 Triangles from Angles (cont
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LAB 3.3 Walking Convex Polygons (co
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LAB 3.4 Regular Polygons and Stars
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LAB 3.5 Walking Regular Polygons (c
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LAB 3.6 Walking Nonconvex Polygons
Page 63 and 64: LAB 3.8 Sum of the Angles in a Poly
Page 65 and 66: LAB 3.9 Triangulating Polygons (con
Page 67 and 68: 4 Polyominoes Polyominoes are a maj
Page 69 and 70: LAB 4.1 Finding the Polyominoes Nam
Page 71 and 72: LAB 4.2 Polyominoes and Symmetry Na
Page 73 and 74: LAB 4.3 Polyomino Puzzles Name(s) E
Page 75 and 76: LAB 4.4 Family Trees (continued) Na
Page 77 and 78: LAB 4.5 Envelopes (continued) Name(
Page 79 and 80: LAB 4.7 Minimum Covers Name(s) Equi
Page 81 and 82: LAB 4.9 Polytans Name(s) Equipment:
Page 83 and 84: LAB 4.10 Polyrectangles (continued)
Page 85 and 86: 5 Symmetry Many of the activities i
Page 87 and 88: LAB 5.1 Introduction to Symmetry (c
Page 89 and 90: LAB 5.2 Triangle and Quadrilateral
Page 91 and 92: LAB 5.3 One Mirror Name(s) Equipmen
Page 93 and 94: LAB 5.4 Two Mirrors Name(s) Equipme
Page 95 and 96: LAB 5.4 Two Mirrors (continued) Nam
Page 97 and 98: LAB 5.5 Rotation Symmetry (continue
Page 99 and 100: LAB 5.6 Rotation and Line Symmetry
Page 101 and 102: LAB 5.7 Two Intersecting Lines of S
Page 103 and 104: LAB 5.8 Parallel Lines of Symmetry
Page 105 and 106: Triangles and 6Quadrilaterals The c
Page 107 and 108: LAB 6.1 Noncongruent Triangles (con
Page 109 and 110: LAB 6.2 Walking Parallelograms (con
Page 111 and 112: LAB 6.4 Making Quadrilaterals from
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Page 117 and 118: LAB 7.2 Tiling with Pattern Blocks
Page 119 and 120: LAB 7.4 Tiling with Regular Polygon
Page 121 and 122: 8 Perimeter and Area This section s
Page 123 and 124: LAB 8.1 Polyomino Perimeter and Are
Page 125 and 126: LAB 8.2 Minimizing Perimeter Name(s
Page 127 and 128: LAB 8.3 A Formula for Polyomino Per
Page 129 and 130: LAB 8.4 Geoboard Area Name(s) Equip
Page 131 and 132: LAB 8.5 Geoboard Squares Name(s) Eq
Page 133 and 134: LAB 8.6 Pick’s Formula (continued
Page 135 and 136: Distance and 9Square Root This chap
Page 137 and 138: LAB 9.1 Taxicab Versus Euclidean Di
Page 139 and 140: LAB 9.2 The Pythagorean Theorem Nam
Page 141 and 142: LAB 9.3 Simplifying Radicals Name(s
Page 143 and 144: LAB 9.4 Distance from the Origin Na
Page 145 and 146: LAB 9.6 Taxicab Geometry Name(s) Eq
Page 147 and 148: 10 Similarity and Scaling Similarit
Page 149 and 150: LAB 10.1 Scaling on the Geoboard Na
Page 151 and 152: LAB 10.2 Similar Rectangles Name(s)
Page 153 and 154: LAB 10.3 Polyomino Blowups Name(s)
Page 155 and 156: LAB 10.3 Polyomino Blowups (continu
Page 157 and 158: LAB 10.5 3-D Blowups Name(s) Equipm
Page 159 and 160: LAB 10.6 Tangram Similarity Name(s)
Page 161 and 162: LAB 10.7 Famous Right Triangles Nam
Page 163 and 164: 11 Angles and Ratios This section p
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LAB 11.1 Angles and Slopes Name(s)
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LAB 11.2 Using Slope Angles Name(s)
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LAB 11.3 Solving Right Triangles Na
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LAB 11.4 Ratios Involving the Hypot
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LAB 11.5 Using the Hypotenuse Ratio
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Trigonometry Reference Sheet The pa
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LAB 11.6 The Unit Circle (continued
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LAB 11.7 Name(s) Perimeters and Are
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LAB 11.8 “” for Regular Polygon
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1 Angles Lab 1.1: Angles Around a P
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For this pattern block hexagon, add
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Theorems: triangle sum theorem, iso
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Answers 1-5. See student work. 6. T
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7. Answers will vary. Discussion An
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compass and straightedge.You may al
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3. Answers will vary. Here are some
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which Question B hints at, is that
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To make an algebra connection, you
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Answers 1. See Polyomino Names Refe
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Answers 1. C. F, N, P D. Only the W
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Lab 4.7: Minimum Covers This lab is
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5. Discussion Answers A. Polyominoe
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information in the relevant cells o
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Discussion Answers A. Pentagons and
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other. Students can make rubber sta
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draw the lines and duplicate them y
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5. Answers will vary. Here are some
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Or students could analyze: • poss
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For Problem 2, make sure the studen
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2 hexagons, 2 triangles 1 hexagon,
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4. P max 2A 2 5. Answers will vary.
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This shows that 2 A⎤ 2n, and sinc
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Lab 8.5: Geoboard Squares Prerequis
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Answers 1. a. 11 b. 10 c. 4 d. 6.4
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3. Discussion Answers A. √5 5 √
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If your students enjoy Problem 6, a
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4. a. 5. a. (0, 0) (6, 0) (0, 0) (0
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congruent to each other and similar
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2. (In the Perimeter table, numbers
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Discussion Answers A. They are the
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Discussion Answers A. Each individu
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3. Triangle c: 1, 3, 2 4. Triangles
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type of slope triangle will work fo
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Discussion Answers A. Answers will
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8. sin 2 cos 2 1. It is the Pythago
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Geometry Labs Geoboard Paper 239 ©
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Geometry Labs 1-Centimeter Grid Pap
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Geometry Labs Circle Geoboard Paper
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To make the radius of this circle e
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Geometry Labs Unit Circles 247 © 1
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GEOMETRY LABS Geometry Labs are han
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