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8.1 C 50 Perimeter 45 40 35 30 25 20 15 10 5 0 5 10 15 20 25 30 35 Area 210 Notes and Answers Geometry Labs © 1999 Henri Picciotto, www.MathEducationPage.org
This shows that 2 A⎤ 2n, and since P 4n, the equation for minimum perimeter is P 2 2 A⎤. Second case: If n 2 A n(n 1), then P 4n + 2.A similar argument to the previous one leads to the inequality 2n 2 A 2n 1. In this case, 2 A⎤ 2n 1, and since P 4n 2, once again the equation for minimum perimeter is P 2 2 A⎤. Answers 1. They alternate between two types of special numbers: perfect squares and the products of consecutive whole numbers. 2. 1, 4, 9, 16, 25 3. 2, 6, 12, 20 4. a. 120 b. 122 c. 120 5. 29 30 870, 30 31 930 6. a. 122 b. 124 c. 118 7. The perimeter increases by 2 whenever a unit square is added to a perfect square or rectangle. Discussion Answers A, B. Squares (and rectangles that are nearly square) are compact and minimize perimeter. In building the spiral, adding a square usually covers two side units from the old perimeter and adds two new side units, which is why the perimeter does not increase. However, when you add a piece to a square or rectangle, you are adding 3 and subtracting 1, so the net effect is to increase the perimeter by 2. C. a. 4n ⎤ b. 4n 2 ⎤ ⎤ ⎤ Lab 8.3: A Formula for Polyomino Perimeter The formula relating area to perimeter and inside dots is a particular case of Pick’s formula, which will be explored in its full generality in Lab 8.6 (Pick’s Formula). The puzzle in Problem 7 is not closely related to the rest of the page, but it can be useful in keeping your fastest students occupied while their classmates continue to work on the previous questions. The wording of Question D is slightly misleading in that there actually is no greatest area for a given number of inside dots, since one can append an inside-dotless tail of any length to a figure. Questions C and D suggest the related questions:What is the greatest number of inside dots for a given area What is the least area for a given number of inside dots These questions are difficult because they are related to the problem of the minimum perimeter for a given area. Answers 1. Perimeter Area Inside dots 2. Answers will vary. 3. You are subtracting 2 from the perimeter. 4. Answers will vary. 5. You are adding 2 to the perimeter. 6. P 2A 2i 2 7. Discussion Answers A. It is the same. B. The first one has fewer inside dots. C. 0 12 9 4 14 9 3 18 11 3 Geometry Labs Notes and Answers 211 © 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
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LAB 4.4 Family Trees (continued) Na
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LAB 4.5 Envelopes (continued) Name(
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LAB 4.7 Minimum Covers Name(s) Equi
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LAB 4.9 Polytans Name(s) Equipment:
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LAB 4.10 Polyrectangles (continued)
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5 Symmetry Many of the activities i
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LAB 5.1 Introduction to Symmetry (c
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LAB 5.2 Triangle and Quadrilateral
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LAB 5.3 One Mirror Name(s) Equipmen
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LAB 5.4 Two Mirrors Name(s) Equipme
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LAB 5.4 Two Mirrors (continued) Nam
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LAB 5.5 Rotation Symmetry (continue
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LAB 5.6 Rotation and Line Symmetry
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LAB 5.7 Two Intersecting Lines of S
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LAB 5.8 Parallel Lines of Symmetry
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Triangles and 6Quadrilaterals The c
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LAB 6.1 Noncongruent Triangles (con
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LAB 6.2 Walking Parallelograms (con
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LAB 6.4 Making Quadrilaterals from
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7 Tiling In this section, we addres
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LAB 7.1 Tiling with Polyominoes Nam
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LAB 7.2 Tiling with Pattern Blocks
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LAB 7.4 Tiling with Regular Polygon
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8 Perimeter and Area This section s
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LAB 8.1 Polyomino Perimeter and Are
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LAB 8.2 Minimizing Perimeter Name(s
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LAB 8.3 A Formula for Polyomino Per
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LAB 8.4 Geoboard Area Name(s) Equip
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LAB 8.5 Geoboard Squares Name(s) Eq
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LAB 8.6 Pick’s Formula (continued
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Distance and 9Square Root This chap
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LAB 9.1 Taxicab Versus Euclidean Di
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LAB 9.2 The Pythagorean Theorem Nam
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LAB 9.3 Simplifying Radicals Name(s
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LAB 9.4 Distance from the Origin Na
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LAB 9.6 Taxicab Geometry Name(s) Eq
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10 Similarity and Scaling Similarit
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LAB 10.1 Scaling on the Geoboard Na
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LAB 10.2 Similar Rectangles Name(s)
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LAB 10.3 Polyomino Blowups Name(s)
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LAB 10.3 Polyomino Blowups (continu
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LAB 10.5 3-D Blowups Name(s) Equipm
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LAB 10.6 Tangram Similarity Name(s)
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LAB 10.7 Famous Right Triangles Nam
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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
Page 175 and 176: Trigonometry Reference Sheet The pa
Page 177 and 178: LAB 11.6 The Unit Circle (continued
Page 179 and 180: LAB 11.7 Name(s) Perimeters and Are
Page 181 and 182: LAB 11.8 “” for Regular Polygon
Page 183 and 184: 1 Angles Lab 1.1: Angles Around a P
Page 185 and 186: For this pattern block hexagon, add
Page 187 and 188: Theorems: triangle sum theorem, iso
Page 189 and 190: Answers 1-5. See student work. 6. T
Page 191 and 192: 7. Answers will vary. Discussion An
Page 193 and 194: compass and straightedge.You may al
Page 195 and 196: 3. Answers will vary. Here are some
Page 197 and 198: which Question B hints at, is that
Page 199 and 200: To make an algebra connection, you
Page 201 and 202: Answers 1. See Polyomino Names Refe
Page 203 and 204: Answers 1. C. F, N, P D. Only the W
Page 205 and 206: Lab 4.7: Minimum Covers This lab is
Page 207 and 208: 5. Discussion Answers A. Polyominoe
Page 209 and 210: information in the relevant cells o
Page 211 and 212: Discussion Answers A. Pentagons and
Page 213 and 214: other. Students can make rubber sta
Page 215 and 216: draw the lines and duplicate them y
Page 217 and 218: 5. Answers will vary. Here are some
Page 219 and 220: Or students could analyze: • poss
Page 221 and 222: For Problem 2, make sure the studen
Page 223 and 224: 2 hexagons, 2 triangles 1 hexagon,
Page 225: 4. P max 2A 2 5. Answers will vary.
Page 229 and 230: Lab 8.5: Geoboard Squares Prerequis
Page 231 and 232: Answers 1. a. 11 b. 10 c. 4 d. 6.4
Page 233 and 234: 3. Discussion Answers A. √5 5 √
Page 235 and 236: If your students enjoy Problem 6, a
Page 237 and 238: 4. a. 5. a. (0, 0) (6, 0) (0, 0) (0
Page 239 and 240: congruent to each other and similar
Page 241 and 242: 2. (In the Perimeter table, numbers
Page 243 and 244: Discussion Answers A. They are the
Page 245 and 246: Discussion Answers A. Each individu
Page 247 and 248: 3. Triangle c: 1, 3, 2 4. Triangles
Page 249 and 250: type of slope triangle will work fo
Page 251 and 252: Discussion Answers A. Answers will
Page 253 and 254: 8. sin 2 cos 2 1. It is the Pythago
Page 255 and 256: Geometry Labs Geoboard Paper 239 ©
Page 257 and 258: Geometry Labs 1-Centimeter Grid Pap
Page 259 and 260: Geometry Labs Circle Geoboard Paper
Page 261 and 262: To make the radius of this circle e
Page 263 and 264: Geometry Labs Unit Circles 247 © 1
Page 266: GEOMETRY LABS Geometry Labs are han
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