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Shape and Space in Geometry
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  Across the Grades
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Shadows are an enduring form of entertainment, from Balinese shadow plays to the finger shadows we have all made when the light is right.

This activity is related to Plot Plans and Silhouettes in that it is about silhouettes. But here, you know the figure and have to imagine the possible shadows as seen from any direction. This is a classic mental rotation problem, with the added twist of your having to project the result as a shadow.

Many people are surprised by the variety of shadows that some figures produce. An interesting discussion may be generated with this question: Can you think of a figure that produces only one shadow? There's only one: a sphere.

across the grades

This activity, as written, is accessible to any grade level simply because it has only yes/no answers. The advantage of technology here is to show how the objects cast the shadows they do.

Thinking about and noticing shadows will help students throughout their school careers as they meet countless practical problems (such as where to park on a hot day) and impractical ones (such as those in math texts about the heights of flagpoles). If students go on in mathematics or computer graphics, they will find these subjects directly related to projective geometry.

standards

The NCTM Standards repeatedly make the point that students need to be exposed to three-dimensional figures, their properties, and their representations in two dimensions.

For example, at grades 3–5, they suggest that all students should

  • investigate and reason about properties of geometric figures;
  • visualize and sketch three-dimensional shapes in two dimensions;
  • describe and draw geometric shapes from recalled mental images.

And at grades 6–8,

  • analyze and understand geometric relationships among two-dimensional and three-dimensional figures;
  • compose and decompose two- and three-dimensional figures in order to solve problems.

Looking at shadows is a simple and less formal way to begin this work in the difficult area of three-dimensional visualization.

resources

Shadows also enter the elementary and middle-grades curriculum through science. Have students record the position of the flagpole shadow throughout a sunny day, or the image of a window (the reverse or negative of a shadow) as it moves across the wall and floor of the classroom.

The science content of an activity such as this is about the earth's rotation and the apparent motion of the sun. But you can extend the activity to include mathematical content as well. Note that the image of the flagpole is always a line segment, whereas the image of the window changes shape. Is it a square? rectangle? parallelogram? rhombus? trapezoid? none of the above?

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