REVIEW
The Geometry Transformation Puzzle is a good tool for developing spatial reasoning and for exploring fractions and a variety of geometric concepts, including size, shape, congruence, similarity, area, perimeter, and the properties of polygons.
This puzzle is especially suitable for independent work in a group setup, since each child can be given a set for which he or she is responsible. Children vary greatly in their spatial abilities and language and most children need ample time to experiment freely with the shapes before they begin more serious investigations.
As a teaching tool, young students will at first think of the shapes literally. With experience, they will see commonalities and begin to develop abstract language for aspects of patterns within their shapes. For example, students may at first make a square simply from two small triangles. Yet eventually they may develop an abstract mental image of a square divided by a diagonal into two triangles, which will enable them to build squares of other sizes from two triangles.
This puzzle can also provide a visual image essential for developing an understanding of fraction algorithms. Many students learn to do examples such as 1⁄2 = ?⁄8 or 1⁄4 + 1⁄8 + 1⁄16 = ? at a purely symbolic level so that if they forget the procedure, they are at a total loss. Students who have had many presymbolic experiences solving problems such as "Find how many small triangles fill the large triangles," or "How much of the full square is covered by a small, a medium, and a large triangle?" will have a solid intuitive foundation on which to build these basic skills and to fall back on if memory fails them.
Young students have an initial tendency to work with others, and to copy one another's work. Yet, even duplicating someone else's solution can expand a student’s experience, develop the ability to recognize similarities and differences, and provide a context for developing language related to geometric ideas. Throughout their investigations, students should be encouraged to talk about their constructions in order to clarify and extend their thinking. For example, students will develop an intuitive feel for angles as they fit corners of the pieces together, and they can be encouraged to think about why some pieces will fit in a given space and others won't. Students can begin to develop a perception of symmetry as they take turns mirroring the pieces across a line placed between them on a mat and can also begin to experience pride in their joint production.
Children of any age who haven't seen this puzzle before are likely to first explore shapes by building objects that look like objects—perhaps a butterfly, a rocket, a face, or a letter of the alphabet. Those with a richer geometric background are likely to impose interesting restrictions on their constructions, choosing to make, for example, a filled-in polygon, such as a square or hexagon, or a symmetric pattern.
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