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LAB 8.2 Minimizing Perimeter (continued) Name(s) 6. What are the minimum perimeters for the following areas a. 925 b. 935 c. 865 In order to understand the <strong>geometry</strong> behind the special role of square and rectangular numbers, you can draw consecutive minimum-area polyominoes by spiraling around an initial square, as shown in the figure below. 7. Continue the process shown in the figure. Describe situations when the perimeter stays the same, and when it changes. Discussion A. Explain why square and rectangular numbers are important in understanding this lab. B. Explain, with the help of figures, why the perimeter of a polyomino whose area is a little less than a given square or rectangular number is the same as the perimeter of the square or rectangle. C. What are the minimum perimeters of polyominoes with the following areas a. n 2 b. n(n 1) 110 Section 8 Perimeter and Area Geometry Labs © 1999 Henri Picciotto, www.MathEducationPage.org
LAB 8.3 A Formula for Polyomino Perimeter Name(s) Equipment: 1-Centimeter Grid Paper, interlocking cubes It is possible to find a formula for the perimeter of a polyomino as a function of its area and of one other variable: the number of inside dots. Inside dots are the points of intersection of grid lines in the interior of the polyomino. For example, the polyomino at right has three inside dots. 1. What are the perimeter, the area, and the number of inside dots for the figures below Experiment on grid paper or with your interlocking cubes to find the formula. Keep track of your experiments in the table. Problems 2–5 can help you organize your research. Perimeter Area Inside dots Perimeter Area Inside dots Geometry Labs Section 8 Perimeter and Area 111 © 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
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LAB 3.8 Sum of the Angles in a Poly
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LAB 3.9 Triangulating Polygons (con
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4 Polyominoes Polyominoes are a maj
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LAB 4.1 Finding the Polyominoes Nam
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LAB 4.2 Polyominoes and Symmetry Na
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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
Page 113 and 114: 7 Tiling In this section, we addres
Page 115 and 116: LAB 7.1 Tiling with Polyominoes Nam
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: LAB 8.2 Minimizing Perimeter Name(s
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
Page 165 and 166: LAB 11.1 Angles and Slopes Name(s)
Page 167 and 168: LAB 11.2 Using Slope Angles Name(s)
Page 169 and 170: LAB 11.3 Solving Right Triangles Na
Page 171 and 172: LAB 11.4 Ratios Involving the Hypot
Page 173 and 174: LAB 11.5 Using the Hypotenuse Ratio
Page 175 and 176: 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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