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Visualizing Dimensions

Children in Froebel's kindergarten played with cubes and with subdivided cubes, squares and subdivided squares, and rods and subdivided rods (Figure 17). Eight small cubes fit together to form a large cube, twice as long, twice as wide, and twice as high. Four square tiles fit together to form a large square, twice as long and twice as wide. Two thin rods form a rod twice as long as the original.

Children at all levels can explore similar exercises. Here is a small cardboard box filled with sand, wrapped in paper, and tied with string. Here is another box — twice as long, twice as wide, and twice as high. How much more string do we need to tie it, or paper to cover it, or sand to fill it? It isn't necessary to have the ability to measure length or area or volume in order to experiment and find the answers: twice as much string, four times as many sheets of paper, eight times as much sand.

Figure 17. Nested cubes, squares, and rods illustrate the fundamental property of doubling factors: they represent the power of 2, depending on dimension.

These perceptions about changes of scale can take place even before the child has much experience with multiplication, and they can reinforce understanding of arithmetic processes. [an error occurred while processing this directive]