Using Manipulatives in the Teaching of K-12 Geometry

Prepared by:

Joseph Malkevitch
Mathematics and Computing Department
York College (CUNY)
Jamaica, NY 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

One of the positive recent developments in K-12 education is that a variety of companies have emerged that sell mathematics manipulatives to teachers and parents. Many of these manipulatives are extremely useful in the area of the teaching of geometry. However, many school systems acquire such manipulatives at considerable cost but rarely provide the training for teachers that is necessary to put these manipulatives to use.

Examples of manipulatives include:

1. D-stix and Zometool

These are rods of different lengths that can be inserted into plastic "nodes" that can be used to make polyhedra. For d-stix the nodes are flexible and the solid angle which forms can change with the model made. For Zometool the rods can only be inserted into the nodes in fixed positions.

2. Origami models

Objects put together or folded from squares of paper.

3. Polydrons and Jovo Toys

These are polygonal panels that can be assembled to form polyhedra.

Polydrons come in two styles: ones where the whole membrane of the polygon is present, and another where only the edge of the polygon is simulated because the "interior" of the polygon has been omitted.) Polydrons now come in the shape of regular 3-gons, 4-gons, 4-gons and 6-gons, as well as right triangles, isosceles triangles, and two types of rhombuses.

Jovo Toy panels are somewhat smaller in size than Polydron panels in sizes with from 3 to 6 sides, and some of the panels have holes.

Membrane panels such as Polydrons and Jovo Toys come in a variety of colors.

4. Rubik's cube

5. Soma cube

The Soma cube is a special cube decomposition puzzle that was invented by Piet Hein. The puzzle is a decomposition of a 3x3x3 cube. It consists of pieces that are 1x1x1 unit cubes which are attached face to face. Since a 3x3x3 has volume 27, which is not divisible by 4 or 5 or 6 one can not decompose this cube into pieces of volume 4, 5, or 6. The pieces that Hein chose have different numbers of unit cubes.

6. Pattern blocks

These are polygonal blocks of various shapes and sizes.

What is the value of using such "toys?"

One of the interesting aspects of studying a particular phenomenon using different kinds of manipulatives is that certain questions seem more likely to come to mind in one format than another.

For example, consider making a cube out of polydrons, out of d-stix, or out origami paper. When you make a cube out of polydrons, since you see only the surface of the cube, it is natural to think about issues related to volume and surface area. Furthermore, the question of determining all the nets of the cube is one that readily suggests itself with a membrane model of the cube. (A net is a plane polygon which when its edges are pasted together edge to edge forms a polyhedron.) In fact, without using something like polydrons, it is quite hard to find all the 11 nets of the cube. (Remember these are the hexominos that fold up to form a cube.) However, rigidity questions about polyhedra are unlikely to occur to one using polydrons.

Manipulates are certainly fun for students and offer up a window to one of the primary aspects of mathematics, studying patterns. However, associated with each manipulate we need to have grade specific activities which include spelled out mathematical goals for the use of these important learning tools.

Here is a sample of questions that are suggested by the use of manipulatives which can be addressed in geometry lessons at different grade levels.

1. If a plane convex polygon has equal length sides and equal interior angles (e.g. a regular polygon) what happens to the interior angle between two consecutive sides as the number of sides of the polygon increases?

2. What happens to the area of a regular polygons with the same edge length as the number of sides of the polygon gets larger and larger?

3. If one opens up a model of a regular tetrahedron (4 equilateral triangular faces) by cutting along its edges to flatten it out into a polygon, how many different results can one get?

4. If one opens up a model of a cube by cutting along its edges to flatten it out into a polygon, how many different results can one get?

5. Is it possible to tile (fill up without holes or overlaps) the plane with equilateral triangles?

6. Is it possible to tile (fill up without holes or overlaps) the plane with squares?

7. Which regular polygons will tile the plane?

8. How many equilateral triangles can be placed to meet at a single point before instead of having a solid angle at the point the polygons flatten out into a plane?

9. How many different convex polyhedra can one make where the faces are all equilateral triangles?

10. Does a rod model of a cube (12 rodes of equal length, 3 at a vertex) flop around when touched or does it show rigidity?

11. Under what conditions do rod models of convex polyhedra not flop around when one handles them (i.e. display rigidity)?

META Question:

At what grade level do you think each of these questions is appropriate?

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