2017 POSB Math and Science Institute
Texas School for the Blind & Visually Impaired
Austin, Texas
April 3-5, 2017
How to Make 3-D Geometry Problems and Concepts
Accessible for Students who are Visually Impaired
Tuesday April 4, 2017
3:45-4:45 PM Breakout Session 9
Presented by
Dr. Aniceta Skowron, Founder, Geometro, Ancaster,
Ontario, Canada
Please do not forget to get the session code at the end of this
session and use this code when you complete your evaluation.
Geometro for Concept Development
Aniceta Skowron, Ph.D.
How to make 3D Geometry problems and concepts accessible
2 parts
Slide 3
1. building pyramids, prisms and more
2. 2. modeling of 3-D problems
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Euler equation
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Diagonals
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Directions
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Labeling and word problem modeling– distances in solids Pythagorean triangles in
solids, volume of solids
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Measuring volume - prisms/pyramids, cylinder/cone
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Nets
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Cross sections
Building Solids
1
Pentagonal Prism
Figure 1 Geometro pentagonal prism
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Has a pentagon & 2 squares in EACH vertex.
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This property can be used as an instruction to build pentagonal prism.
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Pentagonal prism is an example of an Archimedean Solid
Other solids can be built using similar instruction.
For example:
Build a structure that has a pentagon & 3 triangles in each vertex
2017 POSB Math & Science Institute – Skowron, A.
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Pentagonal Anti-prism
Figure 2 Geometro pentagonal anti-prism
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Has a pentagon & 3 triangles in each vertex
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Pentagonal anti-prism is an Archimedean solid
Compare pentagonal prism & pentagonal anti-prism
Figure 3 Two images of Geometro shapes: pentagonal and pentagonal anti-prism.
What are similarities and differenced between the two solids?
Properties of Archimedean solids
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Convex
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Regular polygons for faces
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Not all faces identical – different polygons can be used
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All vertices identical
Figure 4Two images of Geometro shapes: pentagonal and pentagonal anti-prism.
Are pyramids Archidean solids?
Yes
No
Why?
2017 POSB Math & Science Institute – Skowron, A.
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Modeling of concepts and problems
2
Use markers to help count number of faces, vertices and edges in solids
Figure 5 Pentagonal prism with stickers attached to faces
Figure 6 Pentagonal prism with white velcro markers attached to vertices and orange velcro markers attached to edges
Euler’s equation F +V - E = 2
# faces
# vertices
#edges
Hexagonal pyramid
7
7
12
Square prism
6
8
12
Pentagonal pyramid
6
6
10
Pentaonal prism
7
10
15
Triangular pyramid
4
4
6
2017 POSB Math & Science Institute – Skowron, A.
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Orientation of lines
Velcro rods are attached to parallel/perpendicular and skew edges of Geometro cube
Figure 7 Parallel
Figure 8 Perpendicular (normal)
Figure 9 Skew
Orientation of lines
Activity
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Find parallel edges in hexagonal pyramid
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Find parallel edges in pentagonal pyramid
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Make a tetrahedron
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Find all pairs of skew edges in the tetrahedron
Diagonals
Activity
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Show diagonals of pentagonal prism
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How many diagonals are there in hexagonal pyramid?
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Show diagonals in an octahedron
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How many diagonals are there in a square prism?
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Diagonals
1. Definition
2. How many diagonals in each polyhedron?
Figure 10 Four Geometro shapes from right to left: pentagonal prism, hexagonal pyramid, octahedron, square prism.
3. What is the pattern between properties of a polyhedron and the number of diagonals?
4. Where do the diagonals cross?
How to use labels to mark vertices and edges in solids
Figure 11 Three Geometro shapes with letters taped to them, e.g. A, B, C.
Paper clip connectors
Insert a pair of paper clips with velcro (one with hook and one with loop) into ends of a pre-cut
drinking straw. You have a rod with sticky ends, now. The rods can be attached one to
another to form triangles or other polygons.
The rods can also be inserted into solids to show any desired distance in that solid, for example
a diagonal, edge, height of the solid or height of a face. The sticky ends of straws will attach to
the solid’s edges, they can be placed inside or outside the solid.
Figure 12 Images showing how to insert paper clips and rods.
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Slide 20
Triangle made of straws with velcro ends can be inserted into cube. One of the faces of the
cube can be removed and the triangle can be touched.
Figure 13 Three images showing how to insert a triangle into a cube.
A different way to show distances in solids
Insert a paper clip (without velcro) into a pre-cut drinking straw.
Hold the straw in place with a magnet joined to the paper clip
Figure 14 Series of three images showing how to show distances in solids.
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Make a model for one of the following problems
Label all relevant points e.g. vertices, show distances
Slide 23
Problem 1
Cube ABCDA’B’C’D’ has an edge length of 3”. What is the length of the cube’s diagonal?
List all the edges parallel to edge AB.
Problem 2
Figure 15 Hexagonal pyramid
Height of a hexagonal pyramid ABCDEGO is 17” and the length of the hexagon edge is 4.5”.
What are the dimensions of the lateral surface triangles?
Which edge is parallel to BC?
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Compare volume of prism & pyramid
Make square pyramid and square prism (w rectangles for sides)
Fill the pyramid with styrofoam packing peanuts. Transfer the peanuts to the prism. Label the
level of filling using dry erase marker or straws with velcro. What fraction of the prism is filled?
Estimate the volume of the peanuts using one cube inch tray.
Figure 16 Styrofoam peanuts
Figure 17 Cube tray
Figure 18 Pyramid filled with styrofoam peanuts next to a square prism that is empty.
Figure 19 Styrofoam half fills the square prism; the pyramid is empty.
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Compare volume of cone & cylinder
Make cone and cylinder
Fill the cone with styrofoam packing peanuts. Transfer the peanuts to the cylinder. Label the
level of filling. What fraction of the cylinder is filled? Estimate the volume of the peanuts using
one cube inch tray
Figure 20 Styrofoam peanuts
Figure 21 Cube tray
Figure 20 Cone filled with styrofoa peanuts; cylinder is empty.
Figure 21 Cylinder half-filled with styrofoam peanuts; cone is empty.
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Volume of oblique solids
Materials:
Foam squares, 11.2 cmX11.2cm, four colors, stack 1” high of each color
Paper measuring tape (here I use Ikea’s), glued lengthwise (with Scotch tape) to form a narrow
tube, the scale is clearly marked in inches
One paper clip with velcro, bent 90deg
One drinking straw
Four Geometro squares
Volume of oblique solids
Figure 22 Variety of material 11.2 cm long and Geometro squares connected to make the same length.
Make a hanging ruler
Insert bent paper clip into the drinking straw. Insert the straw into tube made of paper
measuring tape such that the straw is fully into the paper tube and only the velcro is outside the
tube.
Figure 23 Image of a hanging ruler.
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Slide 29
Enclose the stack of approximately 4.3inches (five colors) of foam squares in four Geometro
squares. Remove or add a few foam square of the fifth color as needed, so that the cube is full.
Place the cube such that the fifth color is at the bottom of the cube.
Attach the hanging ruler to the front top edge of the cube – on the face without Geometro
square. The ruler has to hang freely.
Figure 24 stack of approximately 4.3inches (five colors) of foam squares in four Geometro squares.
Figure 25 Hanging ruler is attached to the cube.
Volume of oblique solids
In steps, remove the top inch of foam from the stack and incline the vertical sides of Geometro
squares such that the new solid is filled again with the foam. Keep the ruler vertical and
measure the height of the new solid. Discuss how the volume of the oblique prism decreases
with the decrease of the height.
Figure 26 Series of images showing the process described above.
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Oblique prism
Use four drinking straws of equal length with velcro ends and two Geometro squares
Connect the straws to the edges of one square, each straw close to the square’s vertex
Attach the second Geometro square to the other ends of the straws, each straw close to a
vertex. The second square should end up parallel to the first square
Incline the prism by sliding the top square. Discuss how the height of the prism changes
Figure 27 Four images showing the process described above.
Twisted prisms
By rotating the top square relative to the bottom square a series of new solids can be generated
from the initial square prism, as below
Figure 28 A series of images showing how to create twisted prisms.
Nets of 3D solids
Adaptation of Geomero book “Nets of 3D Solids” for vision impaired students resulted in Student
Geometro Workbook Kit, available from APH, http://shop/aph.org
Figure 29 cover of Nets of 3D Solids
Figure 30 Student Geometro Workbook Kit, available from APH
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Nets of 3D solids
11 nets of cube
Figure 31 11 nets of cube
Cross sections of solids
Discover, draw and describe polygons can be formed by cross sectioning a cube
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Use elastics or drinking straws with Velcro ends to show the cross sections
Refer to:
National Museum of Mathematics
Manhattan, NY
www.momath.org
to see how they show cross sections of solids
Not Flat!!!
While using elastics to show cross sections of solids make sure the section is if fact flat. With
elastic it is easy to produce configurations that do not represent a flat section, as shown below
Figure 32 Elastics around a cube.
Cross sections of cube
Figure 33 Cubes with elastics on them to form various cross sections.
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Cross sections of cylinder
Show cross sections of cylinder using elastic bands
Figure 34 Elastics form cross sections of a cylinder using elastics: circle, ellipse, rectangle.
Cross sections of cone
Show cross sections of cone using elastic bands & dry erase marker
Figure 35 Images of cross sections of cones: circle, ellipse, parabola, triangle, hyperbola
Thank you!
Contact [email protected]
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Notes
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Texas School for the Blind & Visually Impaired
Outreach Programs
Figure 36 TSBVI logo.
Figure 37 IDEAs that Work logo and OSEP disclaimer.
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