Geometro: Developing Concepts for Math, Science and O&M with Students who are Visually Impaired October 5, 2012

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1 Texas School for the Blind and Visually Impaired Outreach Programs W. 45 th St. Austin, Texas Geometro: Developing Concepts for Math, Science and O&M with Students who are Visually Impaired October 5, 2012 Presented by Dr. Aniceta Skowron Geometro Developed for Texas School for the Blind & Visually Impaired Outreach Programs

Geometro: Developing Concepts for Math, Science and O&M with Students who are Visually Impaired Aniceta Skowron, Ph.D., Geometro Materials available from Geometro Canada www.

and compare Name the object on the right Figure 1 Prism Figure 2 Anti-prism More anti-prisms Build triangular

3 Geometro: Developing Concepts for Math, Science and O&M with Students who are Visually Impaired Aniceta Skowron, Ph.D., Geometro Materials available from Geometro Canada / info@geometro.net American Printing House for the Blind, Inc. Outline Pyramids, prisms, plus. Geometric concepts Tactile drawings Platonic solids o Archimedean solids Science, art and engineering Build, describe, and compare Name the object on the right Figure 1 Prism Figure 2 Anti-prism More anti-prisms Build triangular anti-prism Figure 3 Pentagonal Build decagonal anti-prism Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

4 Prisms Slide shows a series of prisms and a plus sign + Pyramids Slide shows a series of pyramids. After the first row is the word end?. end Bi-pyramids end + Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach, 201 2

I have 3 different names Figure 4 Trigonal

6 Octahedron Count: edges, vertices, faces

Developing Concepts for Math, Science, and

5 I have 3 different names Figure 4 Trigonal anti-prism Figure 5 Square bi-pyramid Figure 6 Octahedron Count: edges, vertices, faces Show: diagonals, height, orientation of lines, cross section Rotate Reflect Decompose into nets Draw Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach, 201 3

Count Faces Figure 7 A Geometro structure with

Vertices and edges Figure 8 A Geometro structure

and red textured forms attached to the edges.

table below has pictures of various Geometro forms

Solid #faces #vertices #edges F+V-E F V E Geometro:

6 Count Faces Figure 7 A Geometro structure with colored circles affixed to the faces. Vertices and edges Figure 8 A Geometro structure with textured white forms affixed to the vertices and red textured forms attached to the edges. Record the number of faces, vertices and edges The table below has pictures of various Geometro forms under the Solid column. Solid #faces #vertices #edges F+V-E F V E Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach, 201 4

Euler equation for (convex) polyhedral Figure 9 Leonhard Euler was a

He made important discoveries in fields as diverse as infinitesimal calculus

V + F E = 2 Peter Weatherall Figure 10 Picture of Peter Weatherall with his

v=c36h00d7xgs Diagonals Figure 11 Two cubes with rods inserted diagonally

7 Euler equation for (convex) polyhedral Figure 9 Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. V + F E = 2 Peter Weatherall Figure 10 Picture of Peter Weatherall with his guitar. From Mathematicos Fantasticos, CD Diagonals Figure 11 Two cubes with rods inserted diagonally through various corners. Scissors, tape, and a ruler are also shown. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach, 201 5

Height Image below shows a cube with a straw that is 9.

The straw, connectors, scissors, tape and ruler are also included in the slide.

5cm Orientation of lines The slide shows three cubes with red rods attached at

The left-hand cube with rods shows a parallel configuration, the middle a

8 Height Image below shows a cube with a straw that is 9.5 cm inserted through the top and bottom faces. The straw, connectors, scissors, tape and ruler are also included in the slide. straw - 9.5cm Orientation of lines The slide shows three cubes with red rods attached at various edges. The left-hand cube with rods shows a parallel configuration, the middle a perpendicular (normal) configuration, and the right-hand one a skew configuration. The rod devices from Geometro are also shown in the foreground along with the container. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach, 201 6

Find orientation of lines for these solids The slide

Symmetry: Reflection and Rotation The slide shows two

On the left the hands face palm up with the little

On the right the hands are palm down with the thumbs

Rotation Figure 12 Image of a wind turbine.

9 Find orientation of lines for these solids The slide shows a variety of solids and the Geometro rods. Symmetry: Reflection and Rotation The slide shows two pictures of hands. On the left the hands face palm up with the little fingers touching along their edges. On the right the hands are palm down with the thumbs inserted between each other along their inside edges. Rotation Figure 12 Image of a wind turbine. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach, 201 7

Necessary to describe rotation Angle of rotation (the cloth part of the umbrella) Axis of rotation ( the handle of an umbrella) Figure 13 A red umbrella Rotation of pyramids Slide shows a variety of

10 Necessary to describe rotation Angle of rotation (the cloth part of the umbrella) Axis of rotation ( the handle of an umbrella) Figure 13 A red umbrella Rotation of pyramids Slide shows a variety of pyramids, rods placed through some from bottom to top and the Geometro rods and connectors. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach, 201 8

Rotation of the Axis and Rotation of the Angle The slide show a pyramid with a rod

Rotation of prisms The slide show a variety of prisms with Geometro rods to use for

Rotation axis and rotation angle = 90 degrees Slide show a cube with a rod through the

11 Rotation of the Axis and Rotation of the Angle The slide show a pyramid with a rod from bottom to top. Arrows at each end indicate rotation. Rotation of prisms The slide show a variety of prisms with Geometro rods to use for rotation. Rotation axis and rotation angle = 90 degrees Slide show a cube with a rod through the center of the top and bottom. The arrow underneath indicates rotation. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach, 201 9

12 Four rotations of a cube The slide shows a cube. Arrows indicate a rotation of the axis. The number of rotations is equal to Four rotations bring the cube into itself 4-fold rotation symmetry Figure 14 A cube with a rod through the center from bottom to top. Reflection The slide shows various ways to construct a cube using reflection Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

Nets Slide shows a cube and a pyramid with the phrases:

Eleven nets for a cube Slide shows various nets that can form a

of 3-D Solids Student Workbook by Geometro Geometro: Developing

13 Nets Slide shows a cube and a pyramid with the phrases: Decompose to 2-D How many ways? Eleven nets for a cube Slide shows various nets that can form a cube. Tactile Drawings Adaptation Slide shows image of a Nets of 3-D Solids Student Workbook by Geometro Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

appear to converge Figure 15 A train track disappears into a tunnel in the distance.

Perspective Slide shows one-point (top left), two-point (bottom left), and three-point (bottom

14 How to understand tactile drawings? 3-D geometric shapes look/feel very different in 3-d and 2-D In perspective Parallel lines appear to converge Figure 15 A train track disappears into a tunnel in the distance. Figure 16 A perspective drawing of several buildings with grid lines converging at the horizon. Perspective Slide shows one-point (top left), two-point (bottom left), and three-point (bottom right) perspective of a cube (top right). Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

In perspective parallel lines appear to converge

manipulatives The slide shows a variety of solids

Geometro manipulatives used to form a cube around

A red arrow points to the first of the cubes.

around the cube are dismantled to use in drawing.

15 In perspective parallel lines appear to converge Angles change Lengths change New Geometro manipulatives The slide shows a variety of solids that can be made from the Geometro manipulatives How to understand drawing of a cube? The slide shows three images of a cube with Geometro manipulatives used to form a cube around them on the outside. A red arrow points to the first of the cubes. The pictures below show how the manipulatives around the cube are dismantled to use in drawing. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

16 Geometro manipulatives and perspective drawings How to understand broken/dotted lines? Two slides follow showing Geometro manipulatives that fit inside the perspective framework. Some of the manipulatives are shaped liked popcycle sticks and some are squares. Each side has a different color used. How to understand drawing of a pyramid? The slide shows three pyramids with Geometro manipulatives framing each of them. A red arrow is pointing towards the pyramid on the left. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

17 Square pyramid A series of pictures showing Geometro solids and manipulatives and how to draw a pyramid. Watch for adaptation Tactile drawings Manipulatives Figure 17 Image of the Geometro Nets of 3-D Solids Student Workbook. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

Platonic (Regular) solids Figure 18 A stone bust of Plato.

building Figure 19 A photo of a chess board with someone moving a piece.

Geometro tetrahedron A structure in which: 4 triangles meet at EACH vertex Octahedron Figure

18 Platonic (Regular) solids Figure 18 A stone bust of Plato. All faces the same, regular polygons All vertices the same All angles the same Strategy for building Figure 19 A photo of a chess board with someone moving a piece. A structure in which: 3 triangles meet at EACH vertex Tetrahedron Figure 20 Image of a Geometro tetrahedron A structure in which: 4 triangles meet at EACH vertex Octahedron Figure 21 An image of a Geometro octahedron. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

A structure in which: 5 triangles meet at EACH vertex Icosahedron Figure 22 Image of Geometro

A structure in which: 6 triangles meet at EACH vertex Triangular tiling Figure 23 An image of

Figure 24 Image of a Geometro cube (hexahedron) A structure in which: 4 squares meet at EACH

19 A structure in which: 5 triangles meet at EACH vertex Icosahedron Figure 22 Image of Geometro Icosahedron. A structure in which: 6 triangles meet at EACH vertex Triangular tiling Figure 23 An image of a Geometro showing triangular tiling A structure in which: 3 squares meet at EACH vertex Figure 24 Image of a Geometro cube (hexahedron) A structure in which: 4 squares meet at EACH vertex Figure 25 Image of a Geometro showing square tiling Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

A structure in which: 3 pentagons meet at EACH vertex Dodecahedron Figure 26 Image of a Geometro dodecahedron A structure in which: 3 hexagons meet at EACH vertex Figure 27 Image of Geometro showing

20 A structure in which: 3 pentagons meet at EACH vertex Dodecahedron Figure 26 Image of a Geometro dodecahedron A structure in which: 3 hexagons meet at EACH vertex Figure 27 Image of Geometro showing hexagonal tiling Five Platonic Solids Slide shows five Geometro shapes and what they are made up of starting from top left to right bottom: 4 Tetra, 8 Octa, 20 Icosa, 6 Hexa (Cube), 12 Dodeca Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

Shape of Space Image of a table made of Geometro shapes.

and running from top to bottom rows are the letters T,

solids All faces regular polygons, two or more kinds All

Geometro: Developing Concepts for Math, Science, and O&M

21 Shape of Space Image of a table made of Geometro shapes. Across the top of the table are the numbers 3, 4, 5, 6 and running from top to bottom rows are the letters T, S, P, H T S P H Archimedean (semi-regular) solids All faces regular polygons, two or more kinds All vertices the same Figure 28 Drawing of Archimedes. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

22 Summary of truncation Images of Geometro shapes demonstrating truncation Geometro Hoberman sphere and icosidodecahedron Figure 29 Hoberman sphere Figure 30 icosidodecahedron A structure for which one pentagon and two hexagons meet at each vertex Buckyball Soccer ball Figure 31 Buckyball Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

George W. Hart mathematical sculptor http://www.georgehart.com Figure 32 Photo of George W.

Inventing Kindergarten Very interesting and beautifully illustrated book on use of geometry in Froebel

architecture. Figure 33 Image of the book Inventing Kindergarten by Norman Brosterman. Thank you!

23 George W. Hart mathematical sculptor Figure 32 Photo of George W. Hart with two sculpture pieces. Inventing Kindergarten Very interesting and beautifully illustrated book on use of geometry in Froebel kindergarten and the impact the first generation of children attending kindergartens had on modern art and architecture. Figure 33 Image of the book Inventing Kindergarten by Norman Brosterman. Thank you! Contact Figure 34 Image of woman holding a large Geometro shape. Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

24 Texas School for the Blind and Visually Impaired Outreach Programs Figure 35 TSBVI Logo "This project is supported by the U.S. Department of Education, Office of Special Education Programs (OSEP). Opinions expressed herein are those of the authors and do not necessarily represent the position of the U.S. Department of Education. Figure 20 IDEA logo Geometro: Developing Concepts for Math, Science, and O&M for Students who are Visually Impaired Skowron TSBVI Outreach,

Geometro. for accessble 3-D geometry. Aniceta Skowron, Ph.D.

Geometro. for accessble 3-D geometry. Aniceta Skowron, Ph.D. Geometro for accessble 3-D geometry Aniceta Skowron, Ph.D. www.geometro.net aniceta@geometro.net 1. building pyramids, prisms and more 2. accessible 3-D problems Euler equation Diagonals Directions Measuring