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 volume - prisms/pyramids, cylinder/cone Nets Labeling Distances in solids Cross sections
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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
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 has a pentagon & 3 triangles in each vertex Pentagonal anti-prism is an Archimedean solid
Compare pentagonal prism & pentagonal anti-prism What are similarities and differenced between the two solids?
Properties of Archimedean solids Convex Regular polygons for faces Not all faces identical different polygons can be used All vertices identical
Are pyramids Archimedean solids?? Why?
part accessble 3-D concepts
Use markers to help count number of faces, vertices and edges in solids Pentagonal prism with stickers attached to faces Pentagonal prism with white velcro markers attached to vertices and orange velcro markers attached to edges
Euler s equation Hexagonal pyramid Square prism Pentagonal pyramid Pentagonal prism Triangular pyramid F + V - E = # faces # vertices # edges 7 7 12 6 8 12 6 6 10 7 10 15 4 4 6 2 2
orientation of lines parallel perpendicular (normal) skew Velcro rods are attached to parallel/perpendicular and skew edges of Geometro cube
orientation of lines Activity: Find parallel edges in hexagonal pyramd Find parallel edges in pentagonal pyramid Make a tetrahedron Find all pairs of skew edges in the tetrahedron
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? 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
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 sold s edges, they can be placed inside or outside the solid.
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.
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.
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.
Volume of oblique solids 11.2 cm
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.
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.
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.
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 geometro www.aph.org
Nets of 3D solids 11 nets of cube
Cross sections of solids Discover, draw and describe polygons can be formed by cross sectioning a cube Use elastics or pipe cleaners to show the cross sections
Refer to: National Museum of Mathematics Manhattan, NY www.momath.org Ring of Fire to see how they show cross sections of solids
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
Cross sections of cube
cross sections of cylinder Show cross sections of cylinder using elastic bands circle ellipse rectangle
cross sections of cone Show cross sections of cone using elastic bands & dry erase marker circle ellipse parablola triangle hyperbola
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