Enjoy Math! (English) / ART TOWER MITO

1-6-8 Goken-cho, Mito-shi, Ibaraki-ken, 310-0063 Japan
Mail to: webstaff@arttowermito.or.jp
Phone: (029)227-8111 / Fax: (029)227-8110

JOHOKU-CHU at Art Tower Mito
Fun Math: Tenacity Through Art!

Let us introduce to you the story of a teacher who thought up a really fun math class involving the physical shapes extracted from the tower of Art Tower Mito (ATM), as well as the students who worked quite hard at the project and thereby experienced how fun math can be.

The focus of this story is Johoku Junior High School (Johoku Chugakko, or "Johoku-chu" for short), which lies adjacent to northwestern Mito City, in Johoku Town of Higashi-Ibaraki District.

The school can be contacted at:
Johoku-cho-ritsu Johoku Chugakko
Shimo-Aoyama 10, Johoku-machi, Higashi-Ibaraki-gun,
Ibaraki-ken 311-4304 Japan
Tel: +81 29-288-2025
FAX: +81 29-288-2042

Photo credits: Johoku Junior High School (classroom scenes)

Building a Model of the ATM Tower

#3 Class, 1st Year (7th Grade), Johoku-cho-ritsu Johoku Junior High School (April 1999 to March 2000)
Students (34):
Mari Haneishi, Yukari Hoshi, Satomi Iida, Mizuho Itane, Tomomi Kawai, Maki Kobayashi, Sanae Koibuchi, Taiki Koibuchi, Manami Kono, Kenji Kubota, Keiko Kuribayashi, Toru Matsushita, Yuka Matsuzaki, Satoshi Midorikawa, Ayaka Mimura, Kazuya Miyata, Mao Morishima, Takashi Morishima, Manami Nakamura, Hikaru Nakayama, Kenji Nemoto, Mizue Nishi, Atsushi Onuki, Ai Sasaki, Chikako Sonobe, Katsuyuki Sugita, Mitsumasa Takada, Kazumi Takayasu, Yutaro Tominaga, Atsushi Tomita, Yuji Ueda, Yumi Watahiki, Ayaka Yasu, Naoko Yoshimi

Report written by:
Takashi Otsu, teacher-in-charge, #3 Class, 1st Year (7th Grade), Johoku-cho-ritsu Johoku Junior High School
Textbook used:
"Junior High School Mathematics I, New Version" (Dai-Nippon Tosho)
(4 Dai-Nippon Mathematics 709) pp. 153-55

In the last part of February 2000, the #3 Class of first-year students (7th graders) at Johoku Junior High School took part in a class making a model of the ATM tower. The class was planned as a lesson in the unit on "Spatial Figures" in the first-year math class, aiming to fulfill the following goals:

1. Subject Matter: "Let's Make a 3-D Model of ATM Tower"

2. Goals

a. Understand the characteristics of polyhedrons and regular polyhedrons.
b. Understand folding diagrams and the relationship between surfaces, edges, and corners of polyhedrons.
c. Enjoy the beauty of regular polyhedrons and other shapes, thus developing a stronger interest in spatial figures.

3. Student Awareness

Before the project began, I made a survey of how well the students knew about Art Tower Mito and the tower standing in its Plaza. The results are as follows (32 students surveyed in #1 Class, 1st Year):
a. Have you heard of Art Tower Mito before? Yes (97%) No (3%)
b. Have you ever used the facilities there? Yes (0%) No (100%)
c. Have you ever seen the ATM Tower? Yes (100%) No (9%)
d. Have you ever climbed the Tower? Yes (38%) No (62%)
Based on those results, I drew up study materials for five hours of classes.

4. Description of Class

[Hour One]

	Using pamphlets from ATM, students are shown studying the buildings and events there. Students learned that various events are held in the ATM Concert Hall, Contemporary Art Gallery, etc. Students are shown listing their thoughts about the ATM Tower seen in the pamphlet. Their opinions include the following: - It has an interesting shape. - There are a lot of equilateral triangles. - It's twisted. - It's squishy. - It's wonderful.
	After that, students were to "study the Tower's shape in more detail over the Internet," so they moved to the personal-computer room. The students then accessed ATM's home page and studied the Tower's shape. They learned there that the Tower was made of 28 regular tetrahedrons, stacked up regularly, and that it was covered by 57 titanium panels in the shape of equilateral triangles. In the course of the investigation, one student asked, "What is a regular tetrahedron?" I told him we would look into it during Hour Two of the project. That is how Hour One ended.
	Regarding the question posed during Hour One about regular tetrahedrons, the students took time in the classroom to study the meaning of the term. They also learned about polyhedrons and other regular polyhedrons besides regular tetrahedrons, using diagrams.

[Hour Two]

*The image at left is a page from that extremely fascinating textbook.

Chapter 6 "Figures of Shapes"
Let's think about special polyhedrons.

Regular polyhedrons are polyhedrons without any dents, whose surfaces are all identical regular polygons, and whose points are each surrounded by the same number of surfaces. There are only five known regular polyhedrons in existence: regular tetrahedrons, regular hexahedrons (cubes), regular octahedrons, regular dodecahedrons (with 12 faces), and regular icosahedrons (with 20 faces). The diagrams are shown in the same order.

Through the courtesy of Dai-Nippon Tosho Co., Inc., we have reproduced p. 154 of the text that was used ("Junior High School Mathematics I, New Version" (4 Dai-Nippon Mathematics 709), approved by the Ministry of Education on January 31, 1996).

[Hour Three]

The class was broken into into small groups, and each made regular tetrahedrons.

Using construction paper, students drew folding diagrams, then used cellophane tape to construct regular tetrahedrons,

As each group was composed of five to six students, each individual had to make five or so regular tetrahedrons, with each group producing 28 of them (the same number as found in the ATM Tower).

[Hour Four]

Students are shown stacking the regular tetrahedrons that they made. Judging from the students' expressions, they were serious in building the towers, while also having a good time.

One group was never able to stack the tetrahedrons properly, despite trying several times, while another group asked for help after forgetting how many they had stacked up.

Much more time was needed than expected, and Hour Three ended with the groups' towers only being partially built.

Work continued on the stacking of the regular tetrahedrons to make the tower, a task that began in Hour Three. Cheers were heard from each group as they finished: "Great, we did it!" As they stacked the figures up, the students finally realized something. Namely, stacking the tetrahedrons could also produce a spiral reverse to the one found in the ATM Tower. The spiral in the ATM tower twirls in a clockwise fashion as it goes up. The image at the very bottom shows a counterclockwise spiral. It doesn't matter if the reverse-spiral tower is flipped on its head it will still be reverse. I didn't think that far ahead, so I was surprised, upon learning that, that there was a "special way of stacking." About half of the groups ended up making reverse spirals (see third image on the left).

* The members of the group making a reverse spiral appear to have slightly disappointed expressions on their faces. However, yours is not a failure but a "discovery," so have more confidence in yourselves! From before the time the Tower was completed, the staff at ATM had never been aware of reverse spirals, so this was a fresh discovery. I believe that the experience of this discovery will give you great strength in your life to come.

Thereafter, each group counted the number of faces on their towers, and their calculations each came up with the number 58. At first, the students counted each individual face, but they had a very difficult time remembering where they had already counted as they turned their towers around. I asked the students whether or not there might not be a better way to count, and advised them to look into the relation between the number of faces that were hidden as they were building the towers, versus the number of exposed faces. After thinking for a bit, the students realized that the top and bottom tetrahedrons each had three exposed faces, and the ones in the middle only two. Using the equation of (2x3) + (26x2) = 58, the students arrived at the calculation or 58 faces being exposed.

However, ATM's home page mentions that there are only 57 exposed faces, so that left us with the question: "What happened to the other one?" The students conjectured, "Maybe it has something to do with the Tower's foundation." At that point, the class ended. What is the truth?

* Indeed, stringing together 28 regular tetrahedrons in air will result in 58 equilateral-triangle faces being exposed. However, the bottommost face of the regular tetrahedron at the base of the ATM Tower is joined to the roof of the building beneath it. Excluding that face, then, there are 57 faces of titanium panels on the ATM Tower. That is the more detailed explanation.

[Hour Five]
During this class, students were asked to think up their own problems, and then solve them by themselves. The problems they thought of included the following:

(1) Think up a folding diagram to be used in making the tower, and then try building it.
One student did think up such a diagram, and then tried to draw it up, but failed to complete it. He drew up two rows of 19 connected equilateral triangles, and another of 20. Using both inward and outward folds, he attempted to put the three rows together. However, the assembly was exceedingly difficult, and although I took a stab at it, my clumsiness prevented me from doing it well. It was almost impossible to make the tower by folding construction paper, so I gave up halfway through.

* Won't you give it a try?
(Click here to go the "Challenge" page)

(2) Try making the other regular polyhedrons besides the regular tetrahedron.
Most of the students tried their hands at this problem. Using compasses and rulers, they drew folding diagrams on construction paper, and each student, in his or her own fashion, constructed regular polyhedrons. Drawing the pentagons for the icosahedron (20-sided figure) proved to be especially difficult, but thanks to advice from the teacher, the students boldly made their attempts and brilliantly completed their icosahedrons.

(3) Research about quasi-regular polyhedrons (cut-corner icosahedrons). Try building a model.

I told the students how slicing the regular icosahedron in a unique way would produce a shape like that of a soccer ball. Several students then told me they wanted to try making it, so they did. However, owing to the difficulty of drawing the diagram -- composed of both regular pentagons and regular hexagons -- only one student was able to finish it in the end.

(4) Try making a polyhedron using origami.

Students attempted to make polyhedrons using origami, as shown in the textbook. Despite the extreme difficulty of making a 60-sided figure, one student completed it marvelously.

That is a broad overview of what we did in our class. Here are some of the impressions of the students after finishing the lesson:
When I heard we were to build a tower, I thought it would be very difficult and that we couldn't do it. However, having studied the structure of the tower, then making regular tetrahedrons and stacking them up, we were able to do it, and I was very happy when we completed it. My ideas about figures have changed since taking this class.
I learned that many kinds of figures were being used all around me. Although we had our failures, working together to build the tower was really fun, and I was moved when we finished it.
Before, whenever I saw the ATM Tower, I always wondered how they built it, but this class has shown me the Tower's secret. I felt like I wanted to build mysterious figures myself.

This page has been edited from the report drawn up Takashi Otsu, teacher at Johoku Junior High School, thanks to the auspices and cooperation of Yoshio Suzuki, principal of the same school.

As far as the reproduced portions of the textbook are concerned, we would like to express our sincere thanks to the valuable advice and extensive cooperation given by Chikako Yanagisawa, our contact in the Editorial Department of Dai-Nippon Tosho.

Dai-Nippon Tosho has drawn much attention to ATM, having used a photograph of ATM, taken by Tokyo Shimbun, as the frontispiece of its "Junior High School Mathematics 2" textbook (4 Dai-Nippon Mathematics 803, approved by the Ministry of Education on January 31, 1992, and used in Ibaraki Prefecture and elsewhere between 1993 and 1996.

The image is that shown on the cover of the "Junior High School Mathematics 2" textbook. Many people will feel nostalgic when reading the explanation, "An interesting tower has been built by combining regular tetrahedrons."

We are waiting to hear interesting episodes about your math textbooks and classes. Please send them to the following E-mail address:
webstaff@arttowermito.or.jp

In the buildings at ATM are hidden many different shapes and forms. Wonderful art is overflowing at ATM -- both inside and outside those buildings -- be it music, words, or dance movements that strike people's hearts, or artworks that inspire contemplation.

We would love to help arrange a visit by your class or school to ATM, either for a tour or for attendance at an event, among other things. Please feel free to inquire. Also, if you have any inquiries about this Web page, or would like to give us your impressions and opinions, please kindly send an E-mail to the address below. We will pass on any E-mail messages you may have for Johoku Junior High School.

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