About Geometry Gems Overview and motivation These activities are based on a couple of classic types of geometric puzzles, geometric dissections and tangram. The latter is frequently used in elementary classrooms to give children an opportunity to develop spatial reasoning and geometric thinking. The geometric dissections all involve cutting apart a shape to create a different shape from the pieces. Some of the concepts which come into play are congruence and conservation of area. The sides of the pieces which go together must match up and angles must add up correctly to make the new figure. The area of the old shape must be the same as the area of the new shape if all of the pieces cut from the old shape are used to make the new shape. The gems are all classic dissection puzzles from books of mathematical puzzles and recreations. They are conceptually closely linked with tilings. The usual puzzle, which is to take a given polygon and figure out how to cut it up to make some specified polygon, is in most cases remarkably hard to solve. The dissection puzzles on the left panel are similar in spirit to the dissections of the puzzle books but are designed to be problems which elementary students can solve within five or ten minutes. A hinged dissection proof of the Pythagorean theorem (the "Behold!" puzzle) can be put together from Printable Materials and put on the table for children to experiment with. Materials Gems: Cardstock or other lightweight cardboard (Kleenex boxes work nicely and can give a decorative backing to the gems). white school glue Mod-Podge (the sparkle variety makes nice gems) a large artist's brush or a small house paintbrush to spread the Mod-Podge pins (these are optional) colored paper Tangram: Tangram pieces (plastic or cardboard) Printouts of shapes to make from "Printable Materials" Dissections: Printouts of dissection puzzles from "Printable Materials", cut in half "Behold!" Puzzle Printout of the "Behold" puzzle pieces, glue, heavy cardboard. Preparing the Gem Materials Before the math night, print out the gem pieces onto colored paper and cut them apart. I printed three different colors for each gem to offer the students some choice. Keep the octagon/square pieces, the hexagon/star pieces, and the hexagon/diamond pieces in separate piles. Either print the outlines onto cardstock and cut them out, or print them onto paper and trace them onto cardboard to cut out. Doing the Activity When you do the activity, offer the students the octagon/square pieces with a choice of an octagon or a square cardboard backing, hexagon/star pieces with a choice of hexagon or star backing, etc. Allow the students to work out how to put the puzzle pieces together to make the gem. Supervise gluing the pieces onto the cardboard and coating with Mod-Podge, assist if necessary. Glue a pin to the back of the gem (optional). Preparing the Behold Puzzle Print the "Behold" puzzle onto paper and glue the paper onto durable cardboard. Cut out the puzzle parts, cutting around the white circles. Poke holes at the vertices of the triangles inside the white circles. Join the blue corner of the triangle to the blue corner of the irregular shape and the red corner of the triangle to the red corner of the irregular shape using paper fasteners. The triangles can now rotate into two positions. In one position the shape is a blue square next to a red square, with dimensions equal to the two perpendicular sides of the triangle. In the other position the shape is square whose side is the length of the hypotenuse of the triangle. The sum of the squares of the two sides is equal to the square of the hypotenuse. This demonstration can be placed on the table for students to experiment with. About the origins: the 9th century Indian mathematician Bhaskara is supposed to have shown this picture to his colleagues and simply said "Behold!" because this theorem was so clear from his picture. Photo of the Gems and the supplies The Gems Sources and references Websites Geometry: Plane and Fancy by Greg Frederickson illustrates the principle behind the dissections used in the geometry gems. An online java version of the "Behold" puzzle by Greg Frederickson. Geometric dissections on the web, maintained by Greg Frederickson, is a nice collection of dissection websites. "Chemical Dissections" by Ivars Peterson in Science News vol. 163 No. 4, Jan 25, 2003 includes some of the history of geometric dissections, and some current applications in chemistry. Books Several nice dissections are shown in Geometry: An Investigative Approach by Phares G. O'Daffer and Stanley R. Clemens. Mathematical Recreations and Essays by H. S. M. Coxeter and W. Rouse Ball contains a nice collection of dissection puzzles. Henry Dudeney popularized geometric dissection puzzles in the 19th century in his books Amusements in Mathematics and The Canterbury Puzzles and other curious problems. The Penguin Dictionary of Curious and Interesting Geometry by David Wells contains a nice collection of more complicated dissections. Geometry: Plane and Fancy by David Singer provides insight into the connection between tilings and dissections. Dissections: Plane and Fancy by Greg Frederickson shows the tiling which produces the square to octagon dissection. Recreational Problems in Geometric Dissections and How to Solve Them by Harry Lindgren contains another large collection of dissections, some with connections to tiling. Written by Janet A. Dixon 7/11/03 Revised 1/10/06