Key Question What are the geometric properties of a regular tetrahedron, and how do these compare to a bird tetrahedron? Math Properties of polyhedra faces edges vertices Surface area Integrated Processes Observing Comparing and contrasting Relating Paper Two squares per model in one or two colors Additional Materials Bird tetrahedron models Creator Tomoko Fusè Management 1. Because this activity asks students to draw comparisons between the bird tetrahedron and the regular tetrahedron it is very important that both models be made from the same size paper so that the comparisons are valid. Procedure 1. Hand out the folding instructions (pages 33-37) and two squares of paper to each student. Guide Focus Students will fold a regular tetrahedron and compare its properties to those of the bird tetrahedron. students through the construction of the tetrahedron unit step by step. 2. Have students fold the second unit individually, being sure that they made it a mirror image of the first unit. Take the class through the assembly process, giving assistance as needed. 3. When all students have successfully assembled their tetrahedron, hand out the remaining student sheet. Students should also have their bird tetrahedron models on hand. 4. Have students work together in small groups to answer the questions and compare the two models. 5. Close with a time of class discussion where students share the discoveries they made about the properties of their tetrahedron and how these relate to the bird tetrahedron. Discussion 1. How many faces does a regular tetrahedron have? [four] What shape are they? [equilateral triangles] 2. How does this compare to the number and shape of the faces on the bird tetrahedron? [The bird tetrahedron has six faces that are isosceles right triangles.] 3. How many vertices does a regular tetrahedron have? [four] How many edges? [six] How do these values compare to those for the bird tetrahedron? [The bird tetrahedron has five vertices and nine edges.] 4. What was your group s plan for finding the surface area of the tetrahedron? 5. If the base of one face is 3 units and the height is 2.5 units, what is the surface area of the entire tetrahedron? [The surface area of one face is 3.75 units, making the surface area of the entire tetrahedron 15 units 2.] 6. How does this surface area compare to the surface area that you calculated for the bird tetrahedron? [The surface area of the bird tetrahedron is nine units 2.] 7. Were you surprised that the polyhedron with more faces had a smaller surface area? Why or why not? 8. How can you explain this apparent paradox? [While the bird tetrahedron has two more faces than the regular tetrahedron, the surface area of each face is more than two units less than the surface area of each face of the tetrahedron. This accounts for the difference in total surface area of the two polyhedra.] Extensions 1. Have students complete the Equilateral Triangle Exploration. 2. Challenge students to construct a square-based pyramid by using two identical tetrahedron units and the flat square unit from the Tetrahedron Puzzle activity. This can be compared and contrasted with the tetrahedron. 32
1. Fold the square in half vertically and unfold. 2. Fold from the bottom left corner as indicated by the dashed line so that the bottom right corner touches the midline. 3. Fold the right side over so that the two points marked with dots meet as shown. 33
4. Unfold completely and fold the paper horizontally so that the two points marked by dots meet. The horizontal fold should go through the intersection of the two diagonals. 5. a. Fold the top part of the paper down at the point where the bottom edge meets the paper. b. Unfold the bottom half, but leave the top part folded down. a. b. 6. Crease as indicated by each of the dashed lines, bringing the corners in to meet the horizontal midline. 34
7. Fold the top left and bottom right corners where indicated by the dashed lines so that the corners touch the nearest diagonals. Notice that the two new sides formed are parallel to the nearest diagonals. 8. Fold again along the diagonals so that the two sides meet in the center. 9. Flip the paper over and crease where indicated by the dashed lines. 35
10. Repeat steps one through six with the second square. Steps seven through nine will be done as mirror images. Fold the top right and bottom left corners where indicated by the dashed lines so that the corners touch the nearest diagonals. Notice that the two new sides formed are parallel to the nearest diagonals. 11. Fold again along the diagonals so that the two sides meet in the center. 12. Flip the paper over and crease where indicated by the dashed lines. 36
Connect the pieces as shown, folding the units so that each point is inserted into the indicated pocket. You should be left with a regular tetrahedron. Top view Side view 37
Use your completed tetrahedron model to answer the following questions. 1. How many faces does a regular tetrahedron have? What shape are they? 2. How does this compare to the number and shape of the faces on a bird tetrahedron? 3. How many vertices does a regular tetrahedron have? How many edges? How do these values compare to those for the bird tetrahedron? 4. How would you find the surface area of one face of the tetrahedron? of the whole tetrahedron? Describe your plan below. 5. If the base of one face is 3 units and the height is 2.5 units, what is the surface area of the tetrahedron? 6. How does this surface area compare to the surface area you calculated for the bird tetrahedron? 7. Does the difference in these two values surprise you? Why or why not? 38