In Japan, mathematical activities are especially emphasized (MEXT, 2010). Figure 1 shows about mathematical activities. Task

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1 Mishima 449 Planning the mathematics lesson through making 90 System Advertisement -- Focus on mathematical activities and connections in the USA and Japan -- Naoto Mishima, Saitama University, Japan Connections in the USA and Japan In the USA, making connections that is, Linking topics or thinking across grades (CCSS, 2010) is important. In Principles and Standards for School Mathematics (NCTM, 2010, p. 64), the following are emphasized: recognize and use connections among mathematical ideas understand how mathematical ideas interconnect and build on one another to produce coherent whole recognize and apply mathematics in contexts outside of mathematics. At the secondary level understanding of how more than one approach to the same problem can lead to equivalent result is especially emphasized (p. 354). In Japan, mathematical activities are especially emphasized (MEXT, 2010). Figure 1 shows about mathematical activities. Phenomena Mathematization Task Reflection of process Task reposing Consideration Processing Result Application Meaning Enhance language activities in each scene Figure 1. About mathematical activities (MEXT, 2010, p.68) One of the considerations about mathematical activities is mathematization. Thus, educators are concerned with connections between the world of reality and mathematics. The purpose is to illustrate that several approaches to the 90 System Advertisement problem can lead to equivalent results. The method is to show the construction methods of 90 system advertisement. 90 system advertisement and its mathematical model 90 system advertisement(s) are the advertisements in a soccer field on the side of the

2 450 Planning the mathematics lesson through making 90 System Advertisement -- Focus on mathematical activities and connections in the USA and Japan -- goal. When we see them from the main camera fixed at a point in the stadium, they seem to be standing vertically because of hallucination, so I call them imaginary advertisement(s) (See left of Figure 2). Otherwise, when we see them from cameras fixed at other points, they seem to be lying down (See right of Figure 2). Figure system advertisement looks from the main camera (left) and the other cameras (right). The relation between 90 system advertisement and imaginary advertisement is perspective correspondence from the main camera, that is, center of perspectivity. In other words, 90 system advertisement is projected Imaginary advertisement from the main camera to the ground. Becker and Shimada (1997) call areas of thinking related to mathematics, such as solving a problem in a non-mathematical field by applying mathematics and so on, mathematical activities, shown in Figure 3. A mathematical model is placed between the world of reality and mathematics. Figure 3. A model of mathematical activities (Becker & Shimada, 1997, p. 4). 7 th ICMI-East Asia Regional Conference on Mathematics Education

3 Mishima 451 In Figure 4, the main camera is abstracted to a point, the sponsor name is abstracted and Imaginary advertisement is idealized to a rectangle. Figure 4 is the mathematical model of the 90 system advertisement. Figure 4. The mathematical model for the 90 system advertisement The 90 System Advertisement problem and the construction task The problem is to make the flag of England flag as a 90 system advertisement. This advertisement is a 90 system advertisement of the England flag. The construction task is to construct a 90 system advertisement with the following conditions. Point A means the point of the main camera. The foot perpendicular to the ground from point A is point B. I name four points on Imaginary advertisement, top right, bottom right, top left and bottom left as C, D, E and F. The foot perpendicular to straight line FD from point B is point G. The point of intersection of straight line AC and the ground is point H. The point of intersection of straight line AE and the ground is point I. Then, AB = a, BG = b, GF = c, FD = p, CD = q. Figure 5 shows the flag of England. The point of intersection of straight line AK 1 and the ground is point K 1. In the same way, I name as K 2, L 1, L 2, M 1, M 2 because of CK 2 : K 2 K 1 : K 1 D = EL 2 : L 2 L 1 : L 1 F = 3 : 1 : 3, CK 2 = K 1 D = EL 2 = L 1 F = q, K 2 K 1 = L 2 L 1 = q. And, because of CM 2 : M 2 M 1 : M 1 E = DN 2 : N 2 N 1 : N 1 F = 5: 1 :5, CM 2 = M 1 D = DN 2 = N 1 F = p, M 2 M 1 = N 2 N 1 = p

4 452 Planning the mathematics lesson through making 90 System Advertisement -- Focus on mathematical activities and connections in the USA and Japan -- Figure 5. The flag of England Next, I show the construction of quadrilateral FDHI, and using the method of 90 system advertisement for the flag of England. Construction method of by projection chart By parallel-projecting tetrahedron ABHI to the ground and the vertical plane (See left of Figure 6), the projection chart (See right of Figure 6) is given. Figure 6. Tetrahedron ABHI (left) and the construction by projection chart (right) The projection chart can construct quadrilateral FDHI without calculation. It is too difficult to construct a large 90 system advertisement. One needs to construct a reduced projection chart, measure length of FD, DH, FI, IH, IJ, enlarge the lengths to the original size to make the 90 system advertisement. In the same way in the construction of quadrilateral FDHI, the projection chart can make a 90 system advertisement of the flag of England by using the projection chart of tetrahedron ABHI (Figure 7). Figure 7. The making of 90 system advertisement of the flag of England by projection chart Construction method by Pythagorean theorem Focus on DBG. Using the Pythagorean theorem, BD = b 2 + (C + P) 2. In the same way, focus on FBG. By Pythagorean theorem, BF = b 2 + C 2. Focus on ABH. Since 7 th ICMI-East Asia Regional Conference on Mathematics Education

5 Mishima 453 AB // CD, ABH CDH. Because of ratio of similitude is a : q, DH = q b2 +(C+P) 2. In the same way, focus on ABI. Because of ABI EFI, FI = q b 2 +C 2. Since BD : BH = BF : BI(=a q : a), FD // HI. Also, BD : BH = FD : HI, HI = ap. When the foot of the perpendicular to straight line FD from point I is point J, BGF IJF. IJ = bq. Figure 8 shows the constructed quadrilateral FDHI Figure 8. The construction by Pythagorean theorem To make the 90 system advertisement of the England flag, we find the length of DK 1, DK 2, FL 1, FL 2, HM 1, HM 2. Because of DH = q b2 +(C+P) 2, FI = q b 2 +C 2, DK 1 = q b2 +(C+P) 2 a q = ap, HM 1=, DK 2 = q b2 +(C+P) 2, FL 1 = q b 2 +C 2, FL 2 = q b 2 +C 2. And, since HI a q a q a q ap, HM 2= ap. Figure 9 shows the 90 system advertisement of the flag of England.

6 454 Planning the mathematics lesson through making 90 System Advertisement -- Focus on mathematical activities and connections in the USA and Japan -- Figure 9. The making of 90 system advertisement of the flag of England flag by Pythagorean theorem Construction method by vector The conditions of the coordinates of the points that are expressed in three dimensional orthogonal coordinates are the following: A(-c, -b, -a), B(-c,-b,0), C(p, 0, q), D(p, 0, 0), E(0, 0, q), F(0, 0, 0), G(-c, 0, 0). A point of imaginary advertisement is the point S (s x, 0, s y ). The point of intersection of the straight line AS and the ground is the point of T (t x, t y, 0) (Figure 10). Figure 10. The points expressed in three-dimensional orthogonal coordinates Hence, the domain of imaginary advertisement is 0 S x p, 0 S z q 1. On the assumption that AS : AT = u : 1. So, AS = uat FS FA = uft ufa FS = uft + (1 u)fa 0, s z ) = (ut x + (t -1)c, ut y + (t - 1)b, (1 - t)a). b Because of 0 = ut y + (t - 1)b, u =. So, (s x, 0, s z ) = ( bt ct, 0, at ). ty+b t +b t +b Because of 1, 0 bt ct p, 0 at q. t +b t +b So t y b t b c x, t y t x - bp, 0 t y bq. p + c p + c a q Figure 11 shows the constructed quadrilateral FDHI. Figure 11. The construction by vector 7 th ICMI-East Asia Regional Conference on Mathematics Education

7 Mishima 455 For making 90 system advertisement of the flag of England, the conditions of the coordinates of the points that are expressed in three dimensional orthogonal coordinates are the following: K 1 (p, 0, q), K 2 (p, 0, q), L 1 (0, 0, q), L 2 (0, 0, q), M 1 ( p, 0, q), M 2 ( p, 0, q), N 1 ( p, 0, 0), N 2 ( p, 0, 0). Then, the domain of quadrilateral K 1 K 2 L 2 L 1 is 0 s x p, q s z q. Because of (s x, 0, s z ) = ( bt ct, 0, at ), the domain of quadrilateral K t +b t +b 1K 2 L 2 L 1 is t y b t b c x, t y t x p + c - bp, bq t bq y. And, the domain of quadrilateral N 2N 1 M 1 M 2 is p s x p + c a q a q p, 0 s z q. Hence, the domain of quadrilateral N 2 N 1 M 1 M 2 is t y b t x - p+ c bp p + 11c, t y b p+ c t x - bp, 0 t y bq. p + 11c a q Figure 12 shows the 90 system advertisement of the flag of England. Figure 12. The making of 90 system advertisement of the flag of England by vector Planning the lesson with the teaching material I show the 90 System Advertisement problem, the construction task and its three construction methods. Here, with a model of mathematical activities (Becker and Shimada, 1997, p.4) (See Figure 3), I show that the approaches using three construction methods can lead to equivalent results. In this teaching material, (a) the world of reality is the real 90 system advertisement (See Figure 2). (c) The problem is to make the flag of England as a 90 system advertisement. (d) The mathematical model is shown in Figure 4. Then, through process (d) (g), various mathematical models are constructed so that (e) the theory of mathematics and Figure 4 are applied. For instance, when the theory of mathematics is the projection chart, a mathematical model becomes like in Figure 6. In the same process, when the theory of mathematics is the Pythagorean theorem and similar, a mathematical model becomes like in Figure 8, and when the theory of mathematics are vectors and domains, mathematical models become like in Figure 9 and (s x, 0, s z ) = ( bt ct, 0, at ). t +b t +b

8 456 Planning the mathematics lesson through making 90 System Advertisement -- Focus on mathematical activities and connections in the USA and Japan -- (j) The conclusion is that the 90 system advertisement of the flag of England was made using each approach with a theory of mathematics and a mathematical model that follows process (d) (g). (l) The results of each approach are in Figure 7, Figure 9 and Figure 12. At the stage (l), by looking from the main camera, their imaginary advertisement can be seen as the flag of England (Figure 13). 90 system advertisement Imaginary advertisement Setting in the main camera Figure 13. Setting in the main camera, 90 system advertisement of the England flag and its Imaginary advertisement Conclusion The purpose of this study was to illustrate that several approaches to the 90 System Advertisement problem can lead to equivalent results. The method is to show the construction methods for the 90 system advertisement. The finding is that the three approaches, construction methods by projection chart, the Pythagorean theorem and vectors, to the problem can lead to equivalent results. In Japan, projection charts are learned in grade 7, the Pythagorean theorem is learned in grade 9, and vectors are learned in grade 11. Therefore I propose the lesson with this teaching material to be conducted at each grade. This teaching material is also meant to connect between grades. In the future, I will report on the lesson practice that I conducted for grade 10 with this teaching material. References Becker, J. P. & Shimada, S. (1997). The Open-Ended approach: A new proposal for teaching mathematics. Reston, VA: NCTM. CCSS. (2010). Common Core State Standards for Mathematics. MEXT. (2008). Lower Secondary School Teaching Guide for Course of Study: Mathematics. Jikkyo Shuppan. 7 th ICMI-East Asia Regional Conference on Mathematics Education

9 Mishima 457 MEXT. (2010). High School Teaching Guide for Course of Study: Mathematics and Science and Mathematics. Jikkyo Shuppan. Mishima, N. (2014). Development of the teaching mathematical material of 90 system advertisement. Proceedings of the 47th fall study meeting Japan society of mathematical education, p.532. Mishima, N., Takagi, Y., & Matsuzaki, A. (2014). Planning the lesson intend mathematical activities: Focus on making 90 system advertisement. 3rd International Conference of Research on Mathematics and Science Education. Vientiane, Lao: Dong Khamxang Teacher Training College. NCTM. (2000). Principles and standards for school mathematics. Virginia, USA: NCTM. YouTube. (2014, October, 12). Japan vs Cyprus 1-0 (Friendly Match 2014) HD Highlights. Acknowledgement Kirin Company grants a limited permit to use the photo shown in Figure 2. Naoto Mishima Graduate School of Education, Saitama University Shimo-okubo 255, Sakura-ku, Saitama-shi, , JAPAN s14ac203@mail.saitama-u.ac.jp