cardboard, rubber bands, and polyhedron models

Cardboard, Rubber Bands, and Polyhedron ModelsAuthor(s): Patricia F. CampbellSource: The Arithmetic Teacher, Vol. 31, No. 2 (October 1983), pp. 48-52Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41190782 .

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Fig. 1

Cardboard, Rubber Bands, and Polyhedron Models

By Patricia F. Campbell

Young children's preschool years are filled with three-dimensional ob- jects as the children come to understand spatial and topological ideas. These ideas, such as nearness, order, or enclosure, are based on the child's experiences and sensory impressions of shapes and solids. However, too often early school experiences with geometric concepts are limited to plane figures. This is because solids are difficult to represent on paper or the blackboard and young children have difficulty constructing models of three-dimensional figures with cardboard and paste. The "cut and paste" method of construction also produces permanent figures that are difficult to store.

The "cardboard-rubber band" method of constructing three-dimensional models (Stewart 1970) offers a successful alternative for the elementary school. Furthermore, this method allows construction of models that have tunnels, challenging the imagina- tion of older students. Students in the intermediate grades have sufficient dexterity to cut the necessary panels from a pattern. In the younger grades, teachers can cut a set of panels that the children can then assemble into models. The models are easily taken apart and reassembled. One or two sets of panels can be used over and over to construct many different types

Patricia Campbell teaches mathematics educa- tion courses at the University of Maryland, College Park, M D 20742. She is currently doing research on the learning of mathematics by elementary school children.

of solids, making this approach ideal for a learning-center activity. Storage is no problem; the panels from disas- sembled models and the rubber bands will fit into a manila envelope.

Preparing the Panels To start with, triangular-shaped panels are sufficient. Later, you or your students may wish to experiment with panels in the shapes of squares, penta- gons, or hexagons. The key is to make all sides of your panels, whatever the shape of the panels, the same length so you can combine them into a single solid.

The template 1 . Place a piece of lightweight card-

board (either 2-ply bristol board or speech board) beneath the pattern shown in figure 1 .

w

2. Using a sharp-pointed instru- ment such as a geometry compass, make an impression at each of the six labeled corners (А, В, С and Z, F, Z).

3. Remove the cardboard. Using a ruler, draw lines connecting corners A, B, and С as well as corners X, Y, and Z.

4. Cut out the outer shape (along triangle XYZ).

5. Using either the point of a compass or a ballpoint pen, make small (diameter of a pencil point) marking holes at A, B, and С (corners of the inner shape).

This template is now used to construct the panels. Children in the intermediate grades may complete the following steps themselves, but teachers of primary-grade children will need to prepare panels.

The panels 1. Place the template on light-

weight cardboard.

2. Trace the outside of the template.

3. Mark the inner corners through the marking holes with a pencil or ballpoint pen.

4. Remove the template. Cut out around the outer shape you traced.

5. Using a ruler and a pencil or ballpoint pen, connect the three inner corners. Press hard when you draw these lines, as you will fold along these lines later.

6. Using a hand paper punch (1/4- inch diameter for hole), punch three

48 Arithmetic Teacher



Fig. 2

a

Г cut

cut ^ ~

V^ ч cut N - ' b ч - У с d

holes centered over each inner corner (fig. 2a).

7. Using scissors, make two cuts toward the center of each hole. Each cut should be perpendicular to its nearest outer edge (fig. 2b).

8. Panel should now resemble figure 2c.

9. Fold up each tab along the lines connecting the holes. The panel should resemble figure 2d.

Repeat steps 1-9 until the required number of triangular panels for your models are completed. The ease of future model construction is related to the care taken in preparing the panels accurately.

Constructing a Model A model is constructed by laying the panels side-by-side according to a pattern and by placing rubber bands over the tabs of all adjacent panels. Two- inch rubber bands work very well. As more rubber bands are added, the model takes on a three-dimensional shape. When all tabs are connected, the model is complete. Under this method of construction, the tabs are exposed. Students should be remind- ed that these projecting ridges are not part of the abstract forms the model represents.

Patterns for making eight different shapes of increasing complexity are given in figure 3. These models represent the eight deltahedra (plural for deltahedron). Each of these solids consists of panels or faces that are in the shape of equilateral triangles (like the Greek letter, delta) such that no two panels are in the same plane.

Fig.3

А Ж V%A V Tetrahedron Triangular dipyramid V Octahedron

Pentagonal dipyramid Siamese dodecahedron Triaugmented triangular prism

хлХХл '/')w'/' Gyroelongated square dipyramid Icosahedron

October 1983 49



Fig. 4

X Y

A Q D

W Z

Fig. 5 л

' ' ^1

' W=8 cm=^f / V Z

Fig. 6 И/

z Fig. 7

1/"Ц Cube

Dodecahedron

More Patterns Using the procedure just outlined, students can construct templates and panels in the shapes of squares, penta- gons, and hexagons, using guides like those pictured in figures 4, 5, and 6.

The square- and pentagonal-shaped panels can be used to form a cube and a dodecahedron as patterned in figure 7. These shapes - along with the tetrahedron, octahedron, and icosahedron (patterns in fig. 3) - are called Platonic solids. The panels, or faces, of each of these solids consist of one type of a regular polygon (polygons with all sides the same length and all angles the same measure). These are the only five solids that fit this description.

Students enjoy constructing models involving panels of different shapes (fig. 8). If the corners of an octahedron are chopped off, the result is the truncated octahedron. Similarly, the cuboctahedron is a cube with all eight corners chopped off. Still other patterns can be found in Wenninger (1975) or Cundy and Rollett (1954).

For those students who like chal- lenges, the patterns in figure 9 are offered. For these models, the students construct an outer "shell" and the inner "hole" as two separate figures. The "hole" is then covered by the "shell," joining all outside edges. These models represent a type of tun- neled polyhedra known as toroids.

Ideas for Classroom Use Three-dimensional models provide the basis for many interesting activi-




Fig. 8

Triangular prism | J I

V V

Truncated octahedron ' / Cuboctahedron

Fig. 9

Cake pan ><0^. У^ '

Д Cake

lili pan

lös ><0^.

<S> У^ '

V-^7 Shell /' /W Hole ^т-^/у '/

v^

Tortuous tunnel >V ^-^^C^A

Hole Shell i^y |И'/

Pentagonal rotunda ^^"^*

Hole ^ - ^ ^

I I I l I l l ПГП

October] 983 51



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ties in geometry. At the same time, students will be learning to view space as something they can understand, use, and manipulate.

At the primary level, students can begin by learning to place the panels according to a pattern. Patterns made to the actual measure of the panels are necessary at this level. Examining the attributes of constructed models may also lead to a valuable discussion. Questions such as the following may provoke comment:

Are all the panels (or faces) of the model alike or different? What shape(s) are the faces? (If the children have difficulty looking at one face of a complete solid, simply loosen the rubber bands and remove the panel. The panel now provides a two-dimensional pattern that can be traced on paper, yielding a more familiar shape.)

Can you tell one corner (or vertex) apart from any other corner? How could you check your answer?

Does the model look the same or different if you look at it from the side, from the top, or from underneath the model?

How many faces does the model have? (If orientation is a problem, have as many students as necessary place one hand on each face of the model. This establishes a one-to-one correspondence between students and faces, simplifying the counting.) How many corners? How many edges?

As the children become more profi- cient with the models, activities aiding the transition between two-dimensional and three-dimensional representations can be devised. For exam- ple, the patterns may include pictorial images of the solids as well as the layout for the panels (as in figs. 3 and 7). The children learn to refer to the picture to check their construction. In this way the students are gradually developing the ability to identify pictorial representations for geometric solids. Further activities, such as matching patterns with solid models or pictorial representations with patterns, can reinforce this development. This matching using patterns can then be checked as the students assemble the model. Yet another technique is to present drawings of three-dimensional figures from different points of view.

The students then try to hold their models in positions corresponding to the point of view represented in the drawing.

In the middle grades, surface area and volume take on a new meaning with three-dimensional models and patterns. At the same time, concepts such as a closed surface, region exte- rior or interior to a closed surface, or a closed surface as the boundary of an interior region can be explored. By counting and recording the number of vertices (corners), faces (panels), and edges of each of the Platonic solids and the deltahedra, students can discover a relationship among the num- bers of edges, vertices, and faces. This relationship is expressed in Eu- ler's formula V - E + F = 2, where V is the number of vertices, E the number of edges, and F the number of faces.

Hanging completed models from a string not only provides decoration for the classroom but also leads to a discussion of rotational symmetry. If you hang a cube from a vertex (or a face or an edge), will it always look just the same as you rotate it through 360 degrees? What about a tetrahedron or a dodecahedron?

In the elementary school mathematics program, geometry aids children in developing their ability to describe, compare, represent, and relate ob- jects and shapes with which they have experience in their environment. Three-dimensional models can play an important part in this development. The cardboard-rubber band method of construction provides an inexpen- sive means of producing self-stabiliz- ing solids, requiring no adhesives, for the classroom.

Try some of these ideas with your students. You will be surprised by the quality of the results.

References Cundy, Henry M., and A. P. Rollett. Mathe-

matical Models. London: Oxford University Press, 1954. (Available from Creative Publi- cations.)

Stewart, B. M. Adventures among the Toroids. Published by the author, 1970. (Available from B. M. Stewart, 4494 Wausau Road, Okemas, MI 48864.)

Wenninger, Magnus J. Polyhedron Models for the Classroom, 2d ed. Reston, Va.: Na- tional Council of Teachers of Mathematics, 1975. W




cardboard, rubber bands, and polyhedron models

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