planning the mathematics lesson through making ‘90° system...

Mishima

7th ICMI-East Asia Regional Conference on Mathematics Education

11-15 May 2015, Cebu City, Philippines

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.

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

Phenomena

Enhance language activities

in each scene

Application・Meaning

Consideration

・Processing

Result

Task

Mathematization

Reflection of process

Task reposing

Planning the mathematics lesson through making ‘90° System Advertisement’ -- Focus on

mathematical activities and connections in the USA and Japan --



450

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 2. ‘90° 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).

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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 AK1 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

CK2 : K2K1 : K1D = EL2 : L2L1 : L1F = 3 : 1 : 3, CK2 = K1D = EL2 = L1F =

q, K2K1 =

L2L1 =

q. And, because of CM2 : M2M1 : M1E = DN2 : N2N1 : N1F = 5: 1 :5, CM2 =

M1D = DN2 = N1F =

p, M2M1 = N2N1 =

p





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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 = √𝑏2 + (𝐶 + 𝑃)2. In the same

way, focus on △FBG. By Pythagorean theorem, BF = √𝑏2 + 𝐶2. Focus on △ABH. Since

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453

AB // CD, △ABH∽△CDH. Because of ratio of similitude is a : q, DH = 𝑞√𝑏2+(𝐶+𝑃)2

𝑎−𝑞. In

the same way, focus on △ABI. Because of △ABI∽△EFI, FI = 𝑞√𝑏2+𝐶2

𝑎−𝑞. Since BD : BH =

BF : BI(=a – q : a), FD // HI. Also, BD : BH = FD : HI, HI = 𝑎𝑝

𝑎−𝑞. When the foot of the

perpendicular to straight line FD from point I is point J, △BGF∽△IJF. IJ = 𝑏𝑞

𝑎−𝑞. 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

𝑎−𝑞, FI =

𝑞√𝑏2+𝐶2

𝑎−𝑞, DK‟1

＝ 𝑞√𝑏2+(𝐶+𝑃)2

𝑎− 𝑞, DK‟2 ＝

𝑞√𝑏2+(𝐶+𝑃)2

𝑎− 𝑞, FL‟1＝

𝑞√𝑏2+𝐶2

𝑎− 𝑞, FL‟2 ＝

𝑞√𝑏2+𝐶2

𝑎− 𝑞. And, since HI

＝𝑎𝑝

𝑎−𝑞, HM‟1＝

𝑎𝑝

𝑎−𝑞, HM‟2＝

𝑎𝑝

𝑎−𝑞.

Figure 9 shows the „90° system advertisement‟ of the flag of England.





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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 (sx, 0, sy). The point of intersection

of the straight line AS and the ground is the point of T (tx, ty, 0) (Figure 10).

Figure 10. The points expressed in three-dimensional orthogonal coordinates

Hence, the domain of „imaginary advertisement‟ is 0≦Sx≦ p , 0≦ Sz≦ q･･･①. On the

assumption that AS : AT = u : 1. So,

AS ⃗⃗ ⃗⃗ ⃗⃗ = 𝑢AT⃗⃗⃗⃗ ⃗

FS⃗⃗ ⃗⃗ − FA⃗⃗⃗⃗ ⃗ = 𝑢FT⃗⃗⃗⃗ ⃗ − 𝑢FA⃗⃗⃗⃗ ⃗

FS ⃗⃗⃗⃗⃗⃗ ⃗ = 𝑢FT⃗⃗⃗⃗ ⃗ + (1 − 𝑢)FA⃗⃗⃗⃗ ⃗ 0, sz) = (utx + (t -1)c, uty + (t - 1)b, (1 - t)a).

Because of 0 = uty + (t - 1)b, u = 𝑏

𝑡𝑦+𝑏 . So, (sx, 0, sz) = (

𝑏t − 𝑐t

t +𝑏 , 0,

𝑎t

t +𝑏).

Because of ①, 0 ≦ 𝑏t − 𝑐t

t +𝑏 ≦ p , 0 ≦

𝑎t

t +𝑏≦ q.

So ty ≦ 𝑏

𝑐tx, ty ≧

𝑏

𝑝 + 𝑐tx －

𝑏𝑝

𝑝 + 𝑐, 0 ≦ ty ≦

𝑏𝑞

𝑎 − 𝑞.

Figure 11 shows the constructed quadrilateral FDHI.

Figure 11. The construction by vector

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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: K1 (p, 0,

q), K2 (p, 0,

q), L1 (0, 0,

q), L2 (0, 0,

q), M1 (

p, 0, q), M2

(

p, 0, q), N1 (

p, 0, 0), N2 (

p, 0, 0).

Then, the domain of quadrilateral K1K2L2L1 is 0 ≦ sx ≦ p,

q ≦ sz ≦

q. Because of (sx,

0, sz) = (𝑏t − 𝑐t

t +𝑏 , 0,

𝑎t

t +𝑏), the domain of quadrilateral K‟1K‟2L‟2L‟1 is ty ≦

𝑏

𝑐tx, ty ≧

𝑏

𝑝 + 𝑐tx

－ 𝑏𝑝

𝑝 + 𝑐,

𝑏𝑞

𝑎 − 𝑞 ≦ ty ≦

𝑏𝑞

𝑎 − 𝑞. And, the domain of quadrilateral N2N1M1M2 is

p ≦sx ≦

p , 0≦ sz ≦ q. Hence, the domain of quadrilateral N2N1M‟1M‟2 is ty ≦

𝑏

𝑝+ 𝑐tx －

𝑏𝑝

𝑝 + 11𝑐, ty ≧

𝑏

𝑝+ 𝑐tx －

𝑏𝑝

𝑝 + 11𝑐, 0 ≦ ty ≦

𝑏𝑞

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 (sx, 0, sz) = (𝑏t − 𝑐t

t +𝑏 , 0,

𝑎t

t +𝑏).





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(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.

http://www.corestandards.org/

MEXT. (2008). Lower Secondary School Teaching Guide for Course of Study:

Mathematics. Jikkyo Shuppan.

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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. https://www.youtube.com/watch?v=LNAjcbmNdUE.

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, 338-8570, JAPAN

[email protected]

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