Using Origami Activities to Teach Mathematics
Dr. Enrique Ortiz, University of Central Florida, , 2012 NCTM Annual Meeting & Exposition:
Origami Samurai Doll
Objective: Students will construct a geometric shape, and identify geometric shapes and fractional parts of a region.
Procedures:
1. Cut around the perimeter of the square and make the folds as indicated. This is equal to one square unit.
2. Fold back on the dotted line (corner to the center). What shape have you made? Repeat this folding for each corner. After folding the four corners, what shape have you made? What fraction of the square unit is this new shape? How can you express this fraction in exponential form?
3. Fold forward on the dotted line (one corner to the center). What shape have you made? Repeat this folding for each corner. What fraction of the square unit is this new square? How can you express this fraction in exponential form?
4. Fold back on the dotted line (corner to the center). Place a tape on the back of the square (were the words are written), and turn the square over again. What fraction of the square unit is this new square, and express as a fraction? What fraction of the square unit would be represented if you keep folding your paper in this manner, and express in exponential form?
5. Put your finger under one of the small squares and push outward. What shape have you made by doing this?
6. Repeat step 5 with two more small squares. You have made a Momotaro! The square is the head and rest the Samurai outfit.
Explanations for Samurai Doll:
1. Cut around the edges of the square and make the folds as indicated. This is equal to one square unit.
2. Fold back on the dotted line (corner to the center). What shape have you made? Triangle. Repeat this folding for each corner. After folding the four corners, what shape have you made? Square. What fraction of the square unit is this new shape? The fraction of the square unit is ½. How can you express this fraction in exponential form? 1/21 or 2–1.
3. Fold forward on the dotted line (corner to the center). What shape have you made? Pentagon. Repeat this folding for each corner. What fraction of the square unit is this new square? ¼. How can you express this fraction in exponential form? 1/22 or 2–2.
4. Fold back on the dotted line (corner to the center). Place a tape on the back of the square (were the words are written), and turn the square over again. What fraction of the square unit is this new square, and express using exponents? 1/8, 1/23 or 2–1. What fraction of the square unit would be represented if you keep folding your paper in this manner, and express in exponential form? 1/2x or 2–x.
5. Put your finger under one of the small squares and push outward. What shape have you made by doing this? Rectangle.
6. Repeat step 5 with two more small squares. You have made a Momotaro! The square is the head and rest the Samurai outfit.
FOLDING A TRUNCATED TETRAHEDRON
Objective: The students will construct a truncated tetrahedron.
Procedures: Ask students to complete the following steps and answer the questions.
1. Start with an 8-inch diameter circle cut from a piece of paper.
2. Identify the center by folding two diameters. You will be folding the circle so one half of the circumference lies on the other half. When done correctly, you would have what looks like half of a circle. After you fold a diameter, open the paper so you are back to the original circle. Then, repeat the process, again, opening to have a circular paper when you are finished.
3. Open the circle and fold the perpendicular bisector of one of the folded radii (see first part of the figure below). You will be folding the radius so it lies on top of itself and the edge of the circle will pass through the center of the circle.
4. Using one of the points on the circle where the perpendicular bisector of the radius meets the circle, make two more folds to complete the equilateral triangle that has the folded cords as one side. The other end of this fold will be the endpoint of the diameter that held the radius used to make the perpendicular bisector that can be seen. Finally, use the two endpoints from the two partial circles to make the third fold.
5. Fold an isosceles trapezoid by folding one vertex to the midpoint of the opposite side (see figure below). Create an isosceles trapezoid by folding one vertex to the midpoint of the opposite side (locate the midpoint of a side by folding one vertex on top of the other. Note - - To avoid confusion, we are not showing the hidden arcs from the circle or the diameters.
6. Starting with the isosceles trapezoid, make a single fold to get a parallelogram.
7. Make a single fold to get a small, equilateral triangle (see second part of the figure above). Starting with the parallelogram, make a single fold to get an equilateral triangle.
8. Open up the model so a regular tetrahedron is formed (see figure on the right). Open the model so a regular tetrahedron is formed.
9. Flatten out the model to form a two-dimensional, big equilateral triangle (back to end of step 4). Open the model to form a two-dimensional, big equilateral triangle like you had at the end of step 4.
10. Fold the three vertices of the equilateral triangle onto the center of the circle so a regular hexagon is formed.
11. Let the hexagon naturally unfold.
12. Tuck the upper flaps together to a form a model a truncated tetrahedron.
13. Unfold your truncated tetrahedron and this is what you should see (figure below).
Handout for Truncated Tetrahedron with Folding Steps (4-7, and 10)
Adaptations: For more advanced students and if time permits, the following question could be added for further explorations:
1. Ask the following questions after completing step 3:
a. What is the length of the chord?
b. What is the area of the folded region (flap)? Hint: Look at this one after working on the next question.
2. Ask the following question after completing step 4:
a. What is the area of the new folded region [triangle]?
b. What are the angle measures? [of the triangle]?
3. Ask the following question after completing step 5: What is the area of this trapezoid?
4. Ask the following question after completing step 6: What is the area of this parallelogram?
5. Ask the following question after completing step 7: What is the area of this new equilateral triangle?
6. Ask the following questions after completing step 8:
a. What is the surface area of this tetrahedron (triangular pyramid)
b. What is the volume of this shape?
7. Ask the following questions after completing step 10:
a. What is the area of this regular hexagon?
b. What are the angle measures?
c. What are the measures of the angles of these smaller triangles? Possible solution: 60º each.
8. Ask the following question after completing step 12:
a. What is the surface area of the base of the truncated tetrahedron?
b. What is the surface area of the top?
c. What is the surface area of one of the sides?
d. What is the total surface area of the truncated tetrahedron?
e. What is the area of the truncated tetrahedron?
9. Ask students to find and label all segments with their length and all angles with their degree measure.
10. Ask students to make and combine 20 of these truncated tetrahedrons (repeat steps 1-12) to make an icosahedron, and the following questions:
a. What would be the total surface area of the icosahedron?
b. What would be the volume of the icosahedron?
c. What geometric ideas/terms are involved in this activity?
11. As students to find more information regarding the icosahedron: history, background, number of vertices, number of edges. It is one of the Greek solids.
12. What would happen if the icosahedron vertices are truncated?
References:
Sobel, M.A., & Maletsky, E.M. (1999). Teaching mathematics: A sourcebook of aids, activities, and strategies (3rd Edition). Needham Heights. MA: Allyn Bacon.
Ortiz, E., Gresham, G.H., and Brumbraugh, D.K. (2008). TAG – Middle Math is it! Lulu.com: http://www.lulu.com/content/4221270.
Explanations for Truncated Tetrahedron: For more advanced students and if time permits, the following question could be added for further explorations:
1. Ask the following questions after completing step 3:
a. What is the length of the chord?
Possible solution: Solve for x:
Then the length of the chord is , which is 1.73 • 4 @ 6.93”.
b. What is the area of the folded region (flap)? Hint: Look at this one after working on the next question. Possible solution: @ (50.26 – 20.78) ÷ 3 @ 9.83 in.2
2. Ask the following question after completing step 4:
a. What is the area of the new folded region [triangle]? Possible solution: = @ 20.78 in2
b. What are the angle measures? [of the triangle]? Possible solution: 180 ÷ 3 = 60º
3. Ask the following question after completing step 5:
What is the area of this trapezoid? Possible solution: A = = @ 15.6 in2. Or find the area of the small equilateral triangle by dividing the area of the triangle in step 4 by 4: = 5.2 in2, then multiply this quantity by 3: 15.6 in.2.
4. Ask the following question after completing step 6: What is the area of this parallelogram? Possible solution: Unfold the big triangle and visually see the area. Notice that the trapezoid and the parallelogram are related in the same way the pattern blocks are related. Multiply 5.2 in2 by 2 = 10.4 in2.
5. Ask the following question after completing step 7: What is the area of this new equilateral triangle? Possible solution: From step 5: 5.2 in2.
6. Ask the following questions after completing step 8:
a. What is the surface area of this tetrahedron (triangular pyramid)? Possible solution: Multiply 5,2 in2 (from step 5 above) by 4 (number of sides of the tetrahedron) = 20.8 in2.
b. What is the volume of this shape? Possible solution: Volume of a pyramid = = @ 6.92 in.3. The height of the pyramid was found by using the Pythagorean theorem: c2 = a2 + b2; where, c = unknown height, a = 3.31 in. (one of the sides of the pyramid), and b = . This gives us that c2 = (3.31 in)2 + (2.33 in.)2 = 10.96 in, and c or h = 3.99 in.
7. Ask the following questions after completing step 10:
a. What is the area of this regular hexagon? Possible solution: There are several ways to calculate the area of this hexagon. One of them is to calculate the area of the smaller triangles that make up the hexagon. This is the area of the triangle in step 4 above divided by 9 possible small triangles = 20.78 in2 ÷ 9 @ 2.24 in2. Then, multiply this number by 6 (number of small triangles forming the hexagon) = 2.24 in2 • 6 @ 13.44 in2.
b. What are the angle measures? Possible solution: The interior angle measures of the hexagon are 120º each. If we fold this hexagon in half to make a smaller isosceles trapezoid, what is the area of these smaller triangles? It is 2.24 in.2.
c. What are the measures of the angles of these smaller triangles? Possible solution: 60º each.
8. Ask the following question after completing step 12:
a. What is the surface area of the base of the truncated tetrahedron? Possible solution: Use area of triangle from step 4 and divide it by 4: 20.78 in2 ÷ 4 @ 5.2 in2.
b. What is the surface area of the top? Possible solution: Use area of triangle from step 4 and divide it by 9: 20.78 in2 ÷ 9 @ 2.31 in2.
c. What is the surface area of one of the sides? Possible solution: 5.2 in2 ÷ 9 (smaller triangles) = 0.57 in2, 0.57 in2 • 5 (number of smaller triangles for one of the sides of the trapezoid) @ 2.89 in2.
d. What is the total surface area of the truncated tetrahedron? Possible solution:
5.2 in2 + 2.31 in2 + 3(2.89 in2) @ 16.18 in2.
e. What is the area of the truncated tetrahedron? Possible solution: Area of the pyramid – Area of truncated top (smaller pyramid) = 6.92 in3 – = 6.92 in3 – = 6.92 in3 – 1.69 in3 @ 5.23 in3.
9. Ask students to find and label all segments with their length and all angles with their degree measure. Allow students to explore and find as many line segments and angles as possible.
10. Ask students to make and combine 20 of these truncated tetrahedrons (repeat steps 1-12) to make an icosahedron, and the following questions:
a. What would be the total surface area of the icosahedron? Possible solution: 20 • Surface area of one of the faces of the pyramid = 20 • (20.78 in2) @ 415.6 in2.
b. What would be the volume of the icosahedron? Possible answer: 20 • Volume of pyramid = 20 • 6.92 in3 @ 138.4 in3.
c. What geometric ideas/terms are involved in this activity? Answer may vary.
11. As students to find more information regarding the icosahedron: history, background, number of vertices, number of edges. It is one of the Greek solids.
12. What would happen if the icosahedron vertices are truncated? Possible solution: It becomes the shape that resembles a soccer ball. Also, Carbon 60 has a shape similar to the icosahedron.