Using Ornaments to Enhance Students’ Proving Skill in Geometry

Joko Suratno

Mathematics Education Study Program, Universitas Khairun, Ternate, Indonesia

Keywords: Ornaments, Proving skill, Interactive geometry software.

Abstract: There are a large number of geometry ornaments that we can find in everyday life. On the other hand, the

use of these ornaments in mathematics learning, especially geometry, was still very limited. Geometric

patterns or ornaments that there can actually be used as a source of learning mathematics that are full of

creativity. Students who learn by means of these ornaments will become more active and increase their

interest in the learning activities. In addition, learning with ornament media will enhance students' creativity

in making new ornaments, making problems, and solving them with various approaches. Ornaments can

also be used as a tool to develop students' skills in proving. Proof is an important part of mathematics. Proof

is a tool for understanding mathematics. However, many students have difficulties to learn proof. Most

students cannot construct a proof. It might be possible to define learning environments for mathematics

classroom that are apt to enhance students' proving competencies. This study is literature review about who

to proof geometry properties through the activity of constructing geometry objects or ornaments using

geometry software. Interactive geometry software (IGS) is also called Dynamic Geometry Environments

(DGEs) is a computer program that allows user to create and manipulate geometric objects. Cabri II Plus is

one example of IGS that is specialized in two-dimensional programs from around 30s of similar software .

This interactive geometry software has several advantages including being able to make learning geometry

more meaningful, can justify a number of concept errors, and the numerical and numerical capabilities of a

computer also provide a rich learning environment so students can experiment. Exploration experience

using this software also has a good effect on students’ mathematical proof. Apart from that, working with

interactive geometry software has several advantages compared to working manually, including in terms of

accuracy and time.

1 INTRODUCTION

Ornaments or symmetrical patterns in geometry

cannot be separated from ethnomathematics.

Ethnomathematics is the mathematics used by

community groups/cultures, such as the community

cities and villages, groups of workers/laborers,

professional groups, children of a certain age,

indigenous people, and many other groups

recognized by the group's general goals and

traditions (D’Ambrosio, 2006). In addition,

ethnomathematics also related to research that

connects mathematics or mathematics education and

its relationship to the social and cultural

backgrounds, namely research that shows how

mathematics is produced, transferred, disseminated

and specialized in various cultural systems (Zhang

and Zhang, 2010).

One study of cultural systems in

ethnomathematics is called the second strand (Vithal

and Skovsmose, 1997). The ethnomathematics study

at the second level discusses mathematics in

traditional culture. Many kinds of traditional cultures

around us are using mathematics. The culture,

among others, may include geometric patterns and

more can be found at various ornaments and

buildings such as ornamental patterns in the mosque

as a place of worship for Muslims.

There are a lot of geometric patterns in Islamic

culture (Abas, 2001). The ornamental patterns are

divided into three groups, namely, Calligraphic,

Arabesque, and space filling patterns. Most

characteristics of these patterns are in the form of

stars and roses. Therefore, geometric patterns or

ornaments in various buildings can be used as a

source of learning mathematics that is full of

creativity.

194

Suratno, J.

Using Ornaments to Enhance Students’ Proving Skill in Geometry.

DOI: 10.5220/0008899201940199

In Proceedings of the 1st International Conference on Teaching and Learning (ICTL 2018), pages 194-199

ISBN: 978-989-758-439-8

Geometric ornaments are a rich source of

creative applications in geometry. Students who

learn by means of these ornaments will become

more active and increase their interest in the material

being taught. In addition, learning with ornamental

media will enhance students' creativity in making

new ornaments, making problems and solving them

with various approaches (Verner, Massarwe and

Bshouty, 2012).

Enhancing students’ proving skill by creating

ornaments using interactive geometry software is the

first step in increasing the study of

ethnomathematics. The final hope in this study is the

further development and use of this tool in

increasing the creativity of both students and

teachers in mathematics learning.

2 DISCUSSION

2.1 Interactive Geometry Software

Interactive Geometry Software (IGS) is also called

Dynamic Geometry Environments (DGEs) is a

computer program that allows user to create and

manipulate geometric objects. IGS can be divided

into two types, namely two-dimensional (2D) and

three-dimensional (3D) programs. Cabri II Plus is an

example of 2D program from around 30 similar

software. The Cabri Geometry was originally

designed at IMAG, a collaborative research

laboratory between the CNRS (National Center for

Scientific Research) and the University of Joseph

Fourier, in Grenoble, France. Jean-Marie Laborde

started this project in 1985 which aims to facilitate

teachers in teaching geometry (Brainville, 2007).

Interactive geometry software has several

advantages including being able to make learning

geometry more meaningful, can justify some

conceptual errors, and the graphic and numerical

capabilities of computers providing a rich learning

environment so students can experiment freely

(Ismail and Kasmin, 2007). This exploration

experience also offers hope in building a strong

spirit of intuition and geometry estimation that is

important in problem solving and theory

development in the branch of mathematics. In

addition, interactive geometry software is better

when compared to manual work because the right

images can be made in a short time.

2.2 Creating Ornaments

2.2.1 Ornaments on Stairs

This first ornament is located on the stairs that

connects the ground floor with the top floor of a

building (Figure 1). The shape of this ornament is

square, which are one main square and two squares

located in the middle. This ornament is the simplest

ornament compared to other ornaments in next

section.

Figure 1.

Figure 2.

The ornament is made by creating square of ABCD

with Regular Polygon tool located in The Lines

Toolbox. Next, creating a midpoint on AB, BC, CD,

and DA using the Midpoint tool. The four midpoints

are named with letters K, L, M, and N, respectively.

The four points are connected using the Polygon tool

to form the KLMN square. The KLMN square is then

rotated as far as 45

with point O as the center of its

rotation with the Rotation tool and forms PQRS.

2.2.2 Garden Fence Ornaments

This ornament can be found on the garden fence

(Figure 3). The shape of this ornament is the same as

the stairs ornament. Therefore, the steps for making

this ornamental frame are the same as the steps for

making stairs ornament. Next, using the Segment

tool, we can create US, BP, CQ, and DR segments,

as shown in Figure 4.

Figure 3.

Figure 4.

Using Ornaments to Enhance Students’ Proving Skill in Geometry

195

The next process is to remove the labels from

each point (these labels will be used to name the

intersection point between the KLMN and PQRS, the

purpose is only to facilitate the illustration of the

ornament manufacturing process). The process of

creating intersection points between KLMN and

PQRS can be done manually or can also be done

simultaneously using the Intersection Point (s) tool.

After this step, then next is naming the intersection

points as shown in Figure 5. Then points A and F, B

and E, C and H, D and G, B and G, C and F, A and

D, and H and E are connected with the Segment tool

so that shown in Figure 6.

Figure 5.

Figure 6.

The coloring of star ornaments located in the

middle of the image cannot be done in this process.

We need the Polygon tool to make the star. The

coloring of contents of an image can be using the fill

tool and the coloring of lines can be using color tool.

The result of the coloring is shown in Figure 7 and

coloring end of the entire ornament shown in Figure

Figure 7

Figure 8

2.2.3 Wall Ornaments I

The next ornaments are one of the two ornaments

that can be found on the walls of the building

(Figure 9). The level of complexity of these

ornaments is higher than the two ornaments

discussed earlier. In addition, this ornament is also

one of the ornaments that can easily be found in

various Islamic buildings. The reason for choosing

this ornament from the second ornament is because

this ornament has a higher level of difficulty

compared to the second ornament. In addition, the

reason for beauty is also used as a reason why this

ornament was chosen. There are any ways can be

done in designing this ornament. One of them is as

follows. The first, we can draw the square used as

the ornament frame using the Regular Polygon tool

and we name with ABCD. If observed, the main

shape of the ornament is an octagon. Therefore, we

can draw an octagon in a square that we made earlier

as shown in Figure 10.

Figure 9.

Figure 10.

The next process is to make the midpoint of each

side with the Midpoint tool and connect the points

by Polygon tool so that it forms a square. The square

is rotated as far as 22.5

with the Rotation tool with

the center of the polygon as the rotation axis. The

results of the rotation later rotated again and so on

until three times. Then connect a certain number of

points with the Segment tool with the intention as

the basis for making the next ornament, as shown in

Figure 11. The color of the octagon is changed to

another color so that the polygons that will be

created later have a different color from the octagon.

After that, a new multi-faceted aspect has been made

whose line thickness has been changed like a pond

in Figure 12.

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Figure 11

Figure 12

Polygon on Figure 12 rotates as far as 45

so that it

looks like Figure 13. Furthermore, the lines and dots

that are not needed are hidden by the Hide/Show

tool and with a few modifications, ornament was

finish. Products in this process are shown in Figure

14.

Figure 13.

Figure 14.

2.2.4 Wall Ornaments II

The following wall ornaments are the most

complicated ornaments patterns (Figure 15). In

general, this pattern has a ten-sided star in the

middle of the ornament and a five -sided star on the

edge of the ornament as a result of the pattern of

lines formed. One way drawing these ornaments is

as follows. The first, drawing the square and

irregular pentagon in it with the Regular Polygon

tool. Then the regular pentagon formed is rotated as

far as 18

with the center of the rotation angle in the

middle of the lot as shown in Figure 16.

Figure 15.

Figure 16.

The next step is to divide the right and left sides

of the square into four equal parts with the Midpoint

tool. After that, draw two parallel lines as shown in

Figure 17 which will be used as the initial pattern.

The two lines are then rotated one by one as far as

four times with the Rotation and Numerical Edit

tool with the rotating axis in the middle of the

polygon. The results of the rotation can be seen in

Figure 18.

Figure 17.

Figure 18.

The color of the two pentagons in the center of

square was converted. The goal is only to

distinguish the many facets of color that will be

made later. After that, many facets are made which

have thickened the size of the lines that look like

Figure 19 with the Polygon tool. Then the many

facets are rotated as far as 36

with Rotation and

Numerical Edit tool with a rotating axis in the

middle of many facets and produces Figure 20.

Using Ornaments to Enhance Students’ Proving Skill in Geometry

197

Figure 19.

Figure 20.

The next step is to draw five -sided stars that are

outside the two many facets that have been made

before. In this process, Segment and Ray The tool is

used to create the star details that are meant by the

choice of thickened green. The results of this process

can be seen in Figure 21. After that, lines and points

that are not needed are hidden with Hide/Show tool.

The final result of this process is shown in Figure

22.

Figure 21.

Figure 22.

2.2.5 Using Ornament to Proof Geometric

Properties

We have created Figure 6 with interactive geometry

software. To teach student who to proof geometry

properties, we can create a task to students to prove

that FC|| GB. To prove that FC || BG we need to

construct a cycle through P and the center in the

center of square.

SQR = (inscribed angles subtend equal arcs)

QDR is an isosceles triangle

DQ = DR.

QDE RCD

(b EQD = = 90

)

DQ = DR dan =

(vertical angles)

Similarly, we get:

SC = CR = RD = DQ = EP = PF = … .

CD = DE = EF = … .

In the , QE = QD, EF = DC

ED FC. Similarly, HA GB. from HA ED,

It follows that FC GB.

3 CONCLUSION

Ornaments that have been created in the discussion

were a small part of the geometric patterns that can

be found in everyday life. The use of these

ornaments in learning mathematics, especially

geometry is still very possible. Geometric patterns or

ornaments that there can actually be used as a source

of learning mathematics will be full of creativity for

students who learn by means of the ornaments will

be more active and increased interest in the material

being taught (Suratno, Ardiana and Tonra, 2018). In

addition, learning with ornaments media will

enhance students' creativity in making new

ornaments as well as looking for a variety of new

ornaments that one with access it via the Internet.

Teacher can use ornament to teach students about

proving. Similar studies related to geometric

ornaments both derived from Islamic culture or other

religions are still very open. Our next task is to study

and design a form of learning activities by utilizing

the results.

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Cabrilog Sas.

D’Ambrosio, U. (2006) Ethnomathematics: Link between

traditions and modernity. Rotterdam: Sense Publisher.

Ismail, Z. and Kasmin, M. K. (2007) ‘Creating Islamic Art

with Interactive Geometry Software’, in 1st

International Malaysian Educational Technology

Convention.

Suratno, J., Ardiana and Tonra, W. S. (2018) ‘Computer-

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10.1088/1742-6596/1028/1/012132.

Verner, I., Massarwe, K. and Bshouty, D. (2012)

‘Fostering Creativity in Mathematics Teaching

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MathematicsEducation.

Vithal, R. and Skovsmose, O. (1997) ‘The End of

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Zhang, W. and Zhang, Q. (2010) ‘Ethnomathematics and

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