geometry-labs

More documents

Recommendations

Info

Plastic SuperTangrams are available commercially, as are some related puzzle books I authored: SuperTangram Activities (2 vols., Creative Publications, 1986). Answers 1–3. Answers 1. 2. Discussion Answers A. Answers will vary. B. Answers will vary. C. The perimeters are 42,6,4 + 22 (seven shapes), and 2 + 42 (five shapes). Lab 4.10: Polyrectangles This is another generalization of the concept of polyomino, with a basic unit that is no longer a square. Another possible extension is the polyiamond (diamond, triamond, and so on), based on an equilateral triangle unit. Polyiamonds have been discussed by Martin Gardner (in his Sixth Book of Mathematical Games from Scientific American, chap. 18 and elsewhere). 4. 190 Notes and Answers Geometry Labs © 1999 Henri Picciotto, www.MathEducationPage.org
5. Discussion Answers A. Polyominoes with a diagonal line of symmetry, such as the tetromino square or the pentominoes V, W, and X, yield only one polyrectangle. B. Each polyrectangle yields one or two polystamps, depending on whether it is mirror symmetric or not. Each polystamp yields one or two one-positional polystamps. C. False.The less symmetrical shapes yield more versions. 5 Symmetry Lab 5.1: Introduction to Symmetry This is a “getting ready” activity. Understanding the meaning of line and rotation symmetry is a prerequisite to all the <strong>labs</strong> in this section. As a way to introduce these ideas, you may lead a discussion of the activity before handing out the sheet. One way to start such a discussion is to write some student-suggested letters on the chalkboard in two categories,“special” and “not special” (corresponding to “line symmetric” and “not line symmetric”), and see whether students can guess which other letters belong in each category. Shapes other than letters can also be used, of course, and the activity can be generalized to rotation symmetry, or made more specific by requiring criteria such as a horizontal line of symmetry or more than one line of symmetry. In Question C, students should have no trouble classifying the figures according to their rotation symmetries, but it can be difficult to explain what is meant by “n-fold” rotation symmetry. One possible answer is that a figure has n-fold rotation symmetry if it looks exactly the same through n different turns, up to 360°.Another answer is that a figure with n-fold rotation symmetry looks unchanged when rotated by an angle of 360°/n. Point out that any figure looks the same when it’s rotated by 360°. If that’s the only rotation that doesn’t change the figure, then we say the figure does not have rotation symmetry. Make sure your students understand that the word fold, in this context, is not related to folding. Students may also be misled into thinking fold refers to reflection symmetry only, especially since this lab mentions folding in Question B. Note that the letters that have rotation symmetry all have half-turn symmetry.You might discuss whether X and O have rotation symmetries that other letters do not. Bulletin board displays can be created based on the ideas in this activity, especially exhibits of discoveries made by students in working Problem 8.You may have an ongoing search, with new discoveries being added. Some students will probably enjoy looking for the longest possible answer in each. Many more examples of letter symmetries like the one in Problem 7 can be found in Scott Kim’s book Inversions (Key Curriculum Press), the definitive book on the subject. For Question F, if you don’t have a dictionary in your classroom, you may ask your students to look up symmetry as part of their homework. Webster’s College Dictionary (Random House) has this first definition:“the correspondence in size, form, and arrangement of parts on opposite sides of a plane, line, or point; regularity of form or arrangement in terms of like, reciprocal, or corresponding parts.” Geometry Labs Notes and Answers 191 © 1999 Henri Picciotto, www.MathEducationPage.org
Page 1:
GEOMETRY Henri Picciotto LABS ACTIV
Page 4 and 5:
Project Editor: Editorial Assistant
Page 7 and 8:
Contents Introduction .............
Page 9 and 10:
9 Distance and Square Root ........
Page 11 and 12:
Introduction About This Book This b
Page 13 and 14:
• Manipulatives can motivate stud
Page 15 and 16:
If you have access to computers, I
Page 17 and 18:
1 Angles An essential foundation of
Page 19 and 20:
LAB 1.1 Angles Around a Point Name(
Page 21 and 22:
LAB 1.2 Angle Measurement (continue
Page 23 and 24:
LAB 1.4 Angles of Pattern Block Pol
Page 25 and 26:
LAB 1.5 Angles in a Triangle Name(s
Page 27 and 28:
LAB 1.6 The Exterior Angle Theorem
Page 29 and 30:
LAB 1.6 The Exterior Angle Theorem
Page 31 and 32:
LAB 1.7 Angles and Triangles in a C
Page 33 and 34:
LAB 1.8 The Intercepted Arc (contin
Page 35 and 36:
LAB 1.10 Soccer Angles Name(s) Equi
Page 37 and 38:
LAB 1.10 Soccer Angles (continued)
Page 39 and 40:
LAB 1.10 Soccer Angles (continued)
Page 41 and 42:
2 Tangrams Tangram puzzles are quit
Page 43 and 44:
LAB 2.1 Meet the Tangrams (continue
Page 45 and 46:
LAB 2.2 Tangram Measurements (conti
Page 47 and 48:
LAB 2.4 Symmetric Polygons Name(s)
Page 49 and 50:
3 Polygons In the first two labs, t
Page 51 and 52:
LAB 3.1 Triangles from Sides (conti
Page 53 and 54:
LAB 3.2 Triangles from Angles (cont
Page 55 and 56:
LAB 3.3 Walking Convex Polygons (co
Page 57 and 58:
LAB 3.4 Regular Polygons and Stars
Page 59 and 60:
LAB 3.5 Walking Regular Polygons (c
Page 61 and 62:
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
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 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
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
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: Lab 4.7: Minimum Covers This lab is
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 and 226: 4. P max 2A 2 5. Answers will vary.
Page 227 and 228: This shows that 2 A⎤ 2n, and sinc
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
show all

geometry-labs

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?