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
• Euler equation
• Diagonals
• Directions
• Labeling and word problem modeling– distances in solids
Pythagorean triangles in solids, volume of solids
• Measuring volume - prisms/pyramids, cylinder/cone
• Nets
• Cross sections
Building Solids
1
Pentagonal Prism
Figure 1 Geometro pentagonal prism
• Has a pentagon & 2 squares in EACH vertex.
• This property can be used as an instruction to build pentagonal
prism.
• Pentagonal prism is an example of an Archimedean Solid
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Other solids can be built using similar instruction.
For example:
Build a structure that has a pentagon & 3 triangles in each vertex
Pentagonal Anti-prism
Figure 2 Geometro pentagonal anti-prism
• Has a pentagon & 3 triangles in each vertex
• Pentagonal anti-prism is an Archimedean solid
Compare pentagonal prism & pentagonal anti-prism
Figure 3 Two images of Geometro shapes: pentagonal
What are similarities and differenced between the two solids?
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Properties of Archimedean solids
• Convex
• Regular polygons for faces
• Not all faces identical – different polygons can be used
• All vertices identical
Figure 4Two images of Geometro shapes: pentagonal and
pentagonal anti-prism.
Are pyramids Archidean solids?
Yes
No
Why?
Modeling of concepts and problems
2
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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
Hexagonal pyramid
Square prism
Pentagonal
pyramid
Pentaonal prism
Triangular pyramid
# faces
7
6
# vertices
7
8
#edges
12
12
6
6
10
7
4
10
4
15
6
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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
• Find parallel edges in hexagonal pyramid
• Find parallel edges in pentagonal pyramid
• Make a tetrahedron
• Find all pairs of skew edges in the tetrahedron
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Diagonals
Activity
• Show diagonals of pentagonal prism
• How many diagonals are there in hexagonal pyramid?
• Show diagonals in an octahedron
• How many diagonals are there in a square prism?
Diagonals
1. Definition
2. How many diagonals in each polyhedron?
Figure 10 Four Geometro shapes from right to left: pentagonal prism, hexagonal
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.
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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.
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.
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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.
Make a model for one of the following problems
Label all relevant points e.g. vertices, show distances
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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.
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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.
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
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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
Nets of 3D solids
11 nets of cube
Figure 31 11 nets of cube
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Cross sections of solids
Discover, draw and describe polygons can be formed by cross
sectioning a cube
• 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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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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