2. To tessellate: to cover a plane with one or more shapes without overlaps or gaps

3. A Regular Tessellation is when only one regular shape is used. A regular shape has congruent sides and angles A Regular Tessellation is when only one regular shape is used. A regular shape has congruent sides and angles There are exactly 3 Regular Tessellations: Square, Equilateral Triangle & Hexagon

4. The grid for each of these tessellations was done in Sketchpad. Then each student colored her creation by hand, except for Mary who chose to do hers in Paint.

17. A Semi-regular tessellations is when two or more regular shapes are used. At each vertex the same arrangement of shapes must exist. A Semi-regular tessellations is when two or more regular shapes are used. At each vertex the same arrangement of shapes must exist. There are exactly 8 semi-regular tessellations

18. Each of these grids was also done in Sketchpad. Then the students colored them in Paint.

19. 4.8.8. Caitlin #1

21. 3.4.6.4. Ayanna #2

22. 3.4.6.3 Jennifer

23. Catherine 3.3.4.3.4. #3

24. 3.3.4.3.4

25. Vana

26. 3.3.3.4.4. Shannon #4

27. 3.6.3.6. Laurie #5

28. #6

29. Anne Marie 3.3.3.3.6 #7

30. 3.3.3.3.6 Nora

31. Tiffany 3.12.12 #8

33. NEON DRAGONS by Elizabeth

34. Laurie

35. Ayanna’s Ocean

36. Tiffany

37. Tiffany Finished by Vana

38. All of the tessellations done in class were by translation only. They are periodic tessellations. For further explanation go to http://www.geocities.com/dottie4math/slides.ppt To learn more about other types go to http://mathforum.org/sum95/suzanne/tess.intro.html To explore aperiodic tessellations scroll down to Penrose on the homepage http://www.funmath.org Continue with art show.

39. is a fractal created by connecting the midpoints of a triangle to form another triangle. The original triangle is now composed of 4 congruent triangles – each of which is similar to the original. The process continues by connecting the midpoints of the remaining triangles – not the center one. On and on it goes!fractal

40. Students created Sierpenski triangles after the study of congruent triangles and after deriving the proof that states – when the midpoints of two sides of a triangle are connected, that segment is parallel to the third side and equal in measure to ½ its length. While they followed the algorithm, we discussed, after each iteration, how the new triangles related in area and perimeter to the original. We created formulas that allowed us to determine each measure after a given number of steps. For Therapy, each student had the freedom to decorate her triangle in a creative fashion.

41. SHANNON

42. Caitlin

43. Tiffany

46. Katie

50. This one was done with Sketchpad. It’s picture was taken with a digital camera as it appeared on the computer screen. The image was then inserted into the slide show.

51. Reverse Coloring

52. Sierpenski on Calculator Whose idea was this???

53. HA-HA-HA-HA-HA HA-HA-HA-HA! HA-HA-HA!

54. Fractals in the Lab

59. Who is Fibonacci? Fibonacci’s sequence of numbers: 1 1 2 3 5 8 13 21... What is a Palindrome? Ex. radar Fibonacci Palindromic Sequence: 8 5 3 2 1 1 1 2 3 5 8 Making a Palindromic Design: Divide the sides of a polygon into divisions equal in length with the above sequence. Connect corresponding points on adjacent or opposite sides. Then color every other region.

63. AND

64. All snowflakes have a hexagonal shape, but we made 7 sided flakes instead. Our hope is that Mother Nature will show us what true snow- flakes look like. (And she sure did!!!) The Christmas tree has a maze inside. As with a deductive proof, it is sometimes easier if you look at the end first and work backwards. The star on top was created by wrapping a strip of paper into a regular pentagon. Before the wrapping began, a wish was written on the back side of the strip.

66. Mother Nature’s Response

69. There are five Platonic Solids. Each is created with exactly one regular polygon.Platonic Solids One of these solids is called the icosahedron. It is formed with 20 equilateral triangles. At each vertex five triangles meet. If each vertex is truncated, the resultant solid is composed of hexagons and pentagons. The name of this solid is the truncated icosahedron which is one of the Archimedian Solids. There are 13 of these solids which are composed of more than one regular shape.Archimedian

81. I have more fun with mine!

82. Sir David Brewster (1781-1868) History of Kaleidoscopes

83. Kaleidoscopes are created by using mirrors to reflect a design. However, with Sketchpad, the following were made by rotating a triangular design. This is a better technique when the creations are activated. Activation can only be seen in Sketchpad, so in this slide show there is only a pretty image to represent each creation. If you have Sketchpad, see how to see the action at the end of this slide show.

84. Anne Marie

85. Ayanna

86. Caitlin

87. Catherine

88. Catie

89. Elizabeth

90. Jennifer

91. Mary

92. Nora

93. Shannon

94. Laurie

95. Katie

96. Exploring the Mandala

97. Vana

98. Caitlin

99. Catherine

101. Ayanna

102. Catie

103. Jennifer

104. Laurie

107. Katie

109. Math Art And for those beautiful kaleidoscopes Remember you must have Sketchpad on your computer!