From Chaotic to Order: Using Chaos Game in Mathematics Teaching *

: Mathematics has a structure based on concepts and operations with a certain and logical order. The discovery of this order is one of the basic elements of doing meaningful mathematics. It is very important to prepare learning environment that allow students to connect and build relationships between mathematical concepts. In this study, “chaos game” which will enable students to build relations among patterns, probability, series and limits has been introduced in detail in the process of obtaining from irregular cases to regular cases. The Chaos game was explained in detail with examples and given some explanations on why the regular shapes formed at the end of the game. Moreover, some tasks that students will be able to use their abilities such as hypothesis, mathematical connections and deduction and to make connection among some mathematical concepts such as probability, measure, patterns and numbers were formed. Reflections from students about these tasks were also included. In this context, the tasks were applied to 44 freshman students who were attending the department of elementary mathematics education in an education faculty of a state university in Central Anatolia region and some reflections from these tasks were examined. Findings showed that the tasks enabled students to use their skills such as hypothesis, observation, mathematical connections and deduction. It was also determined that the tasks enabled students to make practice on some mathematical topics


Introduction
The constructivist approach emphasizes the learner building connections among their prior and new ideas in the process of constructing their own knowledge. According to this approach, the greater the number of connections among prior and new ideas, the better the understanding (Baki, 2008). Skemp (1976) defined understanding as a measure of quality and quantity of connections that a new idea has with existing ideas. He divides understanding into two parts: relational understanding and instrumental understanding. Skemp (1976) defines instrumental understanding as knowledge of rules and procedures used in applying mathematical process without knowing the reasons. For example, students know that 6/10 can be simplified to 3/5. Moreover, they know that 6/10 is equivalent to 3/5, but not understand that 6/10 and 3/5 indicate the same quantities. Skemp (1976) also defines relational understanding as knowing what to do and why in carrying out mathematical process. Relational understanding comprises when the student realizes the properties of mathematical concepts and build connections among these properties and other mathematical concepts in his/her mind. For example, when a student classifies that the square is also a rectangle and the rectangle is also a trapezoid and the parallelogram is also a trapezoid, it indicates that he/she has relational understanding.
Teaching, lacking in relational understanding, causes generally negative attitude and misconceptions for mathematics among students (Van De Walle, Karp & Bay-Williams, 2012). Connection has a key role in the learning of mathematical concepts and it is also one of the basic skills in teaching mathematical concepts (National Council of Teachers of Mathematics [NCTM], 2000). Activities, materials, examples and explanations used by the teacher in the teaching environment have a great role in gaining this skill. For this reason, it is very important to design environments where students can see interrelated relationships about a concept. The objectives to be created for such environments and to be brought to the students are set out in the curriculum. The deficiencies in the curriculum and the developments required by the era cause the renewal and changes in the programs from time to time.
In Turkey, the most radical reform movement in mathematics education programs has been in 2005 (Baki, 2008). As a result of the reform movement in 2005, the understanding adopted for learning and teaching mathematics has changed and the understanding that knowledge is an active product of the individual and knowledge is not independent from the individual has been accepted (Baki, 2008). In the mathematics education programs created during this period, it was emphasized that the topics were related to real life, and that students created their own knowledge based on concrete experiences and intuition (Ministry of National Education [MoNE], 2007). In line with the new understanding adopted in the curriculum, changes have been made in the topics included in the curricula. For example, for the first time in 2005 elementary mathematics curriculum, fractal geometry, a geometry different from Euclidean geometry, was included (MoNE, 2007). Fractal comes from the Latin verb "frangere", which means "irregular, broken, complex" (Manbelbrot, 1982). Intuitively, fractals are symmetric shapes with respect to magnification (Fraboni & Moller, 2008). More mathematically, a fractal is defined as a shape which has the property of self-similarity-that is it consists of smaller copies of itself with magnification (Karakuş & Baki, 2011). In elementary mathematics curriculum, the activities of building fractals as a result of recursive iteration were included. Fractals were also in 2010 secondary school mathematics curriculum (MEB, 2010). In this curriculum, fractals were built by using transformations which were reflection, translation and rotation and finding various patterns in these shapes (Karakuş & Baki, 2011). Thus, by using fractal geometry in both elementary and secondary school mathematics curriculums, it was tried to help students to make a relationship between geometry and real life and to discover similarities and differences with Euclidean geometry by examining different geometries. Similarly, NCTM (1991) suggests that fractals should be included in mathematics curriculum and that students' attitude and interest can be increased and students can make relationships between mathematics and nature. The fact that fractals can be constructed both geometrically and algebraically shows that they are a good application for studying the relationships between geometry and other areas of mathematics. However, in line with the updates made in mathematics curriculums, fractals were not included in both 2013 and 2017 mathematics curriculums. Although fractals are not a learning outcome in existing mathematics curriculums, they have great importance in discovering many mathematical features and employing different mathematical information in this discovery process. In the literature, it is stated that fractals help students to make different relationships both with the concepts in mathematics and with other disciplines and nature (Adams & Aslan-Tutak, 2006;Bolte, 2002;Devaney, 2004;Fraboni & Moller, 2008;Naylor, 1999;Siegrist, Dover & Piccolino, 2009;Vacc, 1999). For example, Fraboni and Moller (2008) state that students can establish relationships between different subjects of mathematics and make various discoveries while examining the Sierpinski triangle and its properties. Sierpinski triangle (Figure 1) can be constructed by using following steps: 1. Start with an equilateral triangle 2. Subdivide it into four smaller congruent equilateral triangles and remove the central triangle 3. Repeat step 2 with each of the remaining smaller triangles infinitely.  Fraboni and Moller (2008) state that a discovery made by students with the Sierpinski triangle may be in the form of new triangles that occur at each iteration step, and determine the relationship between them and the original triangle. In order to determine this relationship, students should use the information "line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side" (Fraboni & Moller, 2008, p. 198). With this information, they will be able to see that the triangles formed in each iteration step are equal and the triangles formed in the previous iteration step are similar to the original triangle. Thus, while students make discoveries about the Sierpinski triangle, they also make relationships for their knowledge of equality, similarity, parallelism and mid-point theorem "line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side". Similarly, Naylor (1999) gave an activity in calculating the perimeter and area of the Sierpinski triangle. Naylor (1999) states that a number pattern in the form of the powers of 3 can be obtained (for the first step is 1, for second step is 3, for third step is 3 2 =9, for forth step is 3 3 =27, etc.) for the number of triangles removed while the Sierpinski triangle is formed. A similar pattern can be obtained for small black triangles formed at each iteration step. In addition, the perimeter of the Sierpinski triangle can be expressed by a divergent sequence of 3(1+3/2+(3/2) 2 +(3/2) 3 +⋯+(3/2) n ). Since this sequence is divergent, the perimeter of the Sierpinski triangle goes to infinity. In contrast, the area of the Sierpinski triangle converges to zero. This activity shows that students make use of number patterns, exponential numbers, sequences and limit concepts when calculating the perimeter and area of the Sierpinski triangle. In addition, the shapes in Euclidean geometry have a static structure. In other words, Euclidean shapes have a certain perimeter and area. On the other hand, the Sierpinski triangle has an infinite increasing perimeter and also an area approaching towards zero. Such extraordinary situations not only attract students 'attention but also raise questions such as "how can a shape be an infinitive perimeter with zero area?" or "Could there be other shapes like this?"

What is Chaos Game?
As in the studies of Fraboni and Moller (2008) and Naylor (1999), fractals are often created as a result of regular iterations. However, fractals can be created with the help of randomly situations. One of these situations that construct a fractal is chaos game. Chaos game is a game that allows students to see that certain patterns and relationships can be found in many situations randomly expressed in their environment. The rule of the game is very simple: Start with three point such as A, B and C at the vertices of an equilateral triangle and pick any point whatsoever in the triangle; this point is z 0 called the seed. Now roll the die. Depending upon which numbers come up, move the seed half the distance to the similarly numbered vertex. The game is played as follows: Roll the die, if the numbers 1 or 2 come up move the seed half distance to vertex A.
Roll the die, if the numbers 3 or 4 come up move the seed half distance to vertex B.
Roll the die, if the numbers 5 or 6 come up move the seed half distance to vertex C.
Repeat this procedure, each time moving the previous point half the distance to the vertex whose number turns up when the die is rolled. For three steps, the game can be played as follows: pick any point whatsoever in the triangle with the A, B and C at vertices. Roll the dice and if 3 comes up, then move the z 0 seed half the distance to the B vertex. Mark this point as z 1 and this point is the new seed. Roll the die again and 1 comes up, then move the z 1 seed half distance to the A vertex. Mark this point as z 2 and this point is the new seed. Roll the die again and 6 comes up, then move the z 2 seed half distance to the C vertex. Mark this point as z 3 (Figure 2).

Figure 2. Orbit of seed in the Chaos Game
Roll the die again and again a sequence of infinite points such as z 0 , z 1 , z 2 , z 3 , … is obtained. The following questions may come to mind about this sequence of points:  Is the repetition process that creates the dots really random?
 Roll the die many hundreds of times and what will be the resulting pattern of points?
Since, when the dice is rolled, the probability of the numbers matched with the corner points is equal, the iteration process that allows the formation of dots in the game of chaos is random. In addition, the location of the first seed is also random.
As described above, what kind of shape will emerge when the chaos game is played is quite interesting. As a result of the randomness of the game and the sequence of dots behaving randomly, it may be thought the resulting image will be a random smear of points or the points will eventually fill the entire triangle. Below are the figures formed by different number of iteration in the Chaos Game ( Figure 3).

Figure 3. Figures formed in different iterations in Chaos Game
According to Figure 3, as the number of repetitions increases, resulting figure is Sierpinski triangle. This is quite unusual and interesting. Because a random geometric result emerges as a result of a random situation.

Why does the Sierpinski triangle arise from the chaos game?
When playing the chaos game, a starting point "seed" is picked. Suppose that the seed z 0 is in the middle of triangle ABC. When roll the die, the new seed z 1 will be in the half distance between the z 0 and one of the three corner points. Suppose, roll the die and 6 comes up. In this case, the z 1 seed close to C vertex and would be in the middle of the 3 small triangles formed in the first iteration of the Sierpinski triangle ( Figure 4). Similarly, the new seed z 2 will be in the half distance between the z 1 and one of the three corner points. Suppose, roll the die and 4 comes up. In this case, the z 2 seed close to B vertex and would be in the middle of the 9 small triangles formed in the second iteration of the Sierpinski triangle ( Figure 5). Again play the game, the new seed z 3 will be in the half distance between the z 2 and one of the three corner points. Suppose, roll the die and 1 comes up. In this case, the z 3 seed close to A vertex and would be in the middle of the 27 small triangles formed in the third iteration of the Sierpinski triangle ( Figure 6). Similarly, the new seed z 4 will be in the middle of the 81 small triangles formed in the third iteration of the Sierpinski triangle. Thus, the z 1 , z 2 , z 3 ,… points will be in the triangles formed in the different iteration of the Sierpinski triangle. The points will lie smaller triangles and these triangles very quickly become microscopic in size. So, the orbit looks like it lies on Sierpinski triangle.
Chaos game which will enable students to build relations among patterns, probability, series and limits has been introduced in detail in the process of obtaining from irregular cases to regular cases (Devaney, 2004). The purpose of this study was to prepare tasks in which students build relations among mathematical concepts and to give reflections from the implementation of these activities. The problems of this study were as follows:  What are the reflections from students about tasks?  When the tasks were implemented, what kind of mathematical relationships do the students build?
The Chaos game was explained in detail with examples and given some explanations on why the regular shapes formed at the end of the game. Moreover, some tasks that students will be able to use their abilities such as hypothesis, mathematical connections and deduction and to make connection among some mathematical concepts such as probability, measure, patterns and numbers were formed. In this context, the tasks were applied to students and some reflections from these tasks were examined.

Method
The research methodology of this study was a case study, since students design activities that will create different associations between mathematical concepts in chaos and examine the associations of elementary school mathematics freshmen students towards these activities. Case studies give the researcher the opportunity to describe the special cases examined with a special focus on a very specific subject or situation, and to explain the cause-effect relationships between the variables (McMillan & Schumacher, 2014).

Sample
The sample of this study has been determined by convenience sampling, which is one of the non-random sampling methods. The reason for choosing the convenience sampling method in the study is that the group to be examined is accessible and practicable due to the limitations in terms of time, money and labor (McMillan & Schumacher, 2014). The participants of this study consisted of 44 freshman students who were attending department of elementary mathematics education in an education faculty of a state university in Central Anatolia region. 36 of these students are women and 8 are men. None of the students have any prior knowledge of the chaos game.

Data Collection
Data were collected from students' written explanations and focus group interviews. Three tasks were formed for chaos game. The aim of the first task was to ensure that students can recognize regular shapes formed at the end of a random process. Thus, students will be created an understanding on chaos and chaotic thinking. The aim of the second task was to make a connection between chaos game and Sierpinski triangle. The aim of the last task was about the probability of the roll a die and construction of Sierpinski triangle. The tasks were prepared taking into account the development steps of a task in the study of Baki (2008). After the tasks were prepared, these tasks were presented to the two academics who were expert about fractals and also completed their doctoral education in the field of mathematics education. Experts have stated that the tasks are generally appropriate, but some minor corrections can be made. For example, it was stated that it would be more appropriate for students to enter their own values in addition to the probability values given in task 3. For this purpose, a place is reserved for students to write and examine their own values under the probability values given in task 3. According to the opinions of the experts, the final arrangements were made in the activities and application started. Before applying the tasks, students were separated into eleven groups and each group consists of four students. During the implementation of the tasks, each group was asked to fill in the activity sheets in line with the guidelines and explain the reasons for the results they obtained. In the second stage of data collection, focus group interviews were conducted with 16 students in 4 groups who participated in the study and volunteered. The purpose of the focus group discussion is to reveal the thoughts of the students that do not appear on the activity sheets, to examine the reasons of the answers they have given and to discuss their answers. The focus of the group discussions is the answers given by the students to the activities. Therefore, the students were asked questions such as, "How did you get this result?", "What did you observe?", "Why did you write this answer?". The aims of the questions are to reveal what kind of thinking processes students have. Focus group interviews were recorded on audio and each interview lasted an average of 30 minutes.

Implementation of the tasks
Before applying the tasks, students were separated into eleven groups and each group consists of four students. While the students were doing the activities as a group work, one of the researchers guided the students by walking between the groups, with clue questions and guidance when the students had difficulties. The researcher who carried out the application is a mathematics educator experienced in fractals. The researcher has many studies in national and international refereed journals about fractals and teaching fractals. The activities were carried out for a total of 4 class hours, 2 class hours per week, for 2 weeks. In the implementation of the activities, a computer laboratory was used and a computer connected to the internet was given to each group. The first activity lasted two hours and students played chaos game with the help of transparent paper and pencil. The reason for playing the game in this way is to make students realize that the movement of the dots is random and the probability of rolling dice is equal. In addition, it is aimed to develop skills such as hypothesis, observation and inference about what kind of shapes will be formed at the end of the game. After playing the game on paper, the students were asked to turn on the computers and they were allowed to play the game on the website http://www.shodor.org/interactivate/activities/TheChaosGame/. Since the website allows more points to be formed in a short time, it provides a more accurate Sierpinski triangle. The second activity lasted one hour and the students were aimed to establish a relationship between the Chaos game and the Sierpinski triangle and to realize how the Sierpinski triangle formed as a result of the movements of the points in the game. In this activity, students are expected to use their skills such as pattern finding, prediction and inference. The last activity lasted 1 lesson and once more http://www.shodor.org/interactivate/activities/TheChaosGame/ website was used. In this activity, different probability states of the numbers in the dice were written in the program on the website and the students observed the shapes that occurred in each probability case and the activity also aimed to determine the conditions of the formation of the Sierpinski triangle as a result of these observations.

Data analysis
Students' explanations for each task were compared and they divided into two categories as right explanations and wrong explanations. Then, each category was examined in its own way and the reflections from the experiences of the students were presented descriptively. For this purpose, direct quotations were excerpted from the explanations of students. The data obtained from the focus group interviews were presented descriptively to support the responses of the students to the tasks. In order to ensure the internal validity, the control of the data obtained from the focus group interviews was done with participant confirmation (Fraenkel & Wallen, 2011). In addition, data triangulations (Cohen, Manion & Morrison, 2000) was made using both the responses to the activities and the data obtained from the focus group interviews. To ensure external validity, quotations from students' answers and focus group interviews are included in the findings section.

Reflections from the "Chaos Game"
This activity is designed to enable students to see that regular shapes can occur at the end of a random process, thereby creating an understanding of chaotic thinking. While the first part of the activity was held in paper-pencil environment, the second part was held in computer environment. Playing the game primarily in paper-pencil environment helps students both see the movement of the dots and realize that the process is random. The randomization of the starting point in the chaos game and the randomness of the numbers in the dice throw causes a perception that a triangle consisting of random points may be formed at the end of the process. At the beginning of the activity all groups stated that, a triangle covered with dots will be formed as a result of the game. For instance, the findings from the focus group interview with Group 1 are presented below: Researcher: In the beginning, when you were playing the game, what kind of shape did you think would occur? Group 1: We thought that a triangle would be formed which was not completely clear. Researcher: Why did you think such a shape would occur? Group1: We marked a random point between points A, B, C. Since we do not know what the number from the dice will be, we first thought that a random shape would be formed. Then we saw that no point overlapped, and all the points remained inside. Therefore, we thought that a triangle filled with dots would form.
When groups put the markings they made on transparent papers (see Figure 7) on top of each other and combined them, they obtained the observations in Table 1.

Figure 7.
A student's markings for points in the chaos game Table 1. Observations of the groups in the chaos game activity Observations f Too many points did not fall where the triangle's center of gravity is.
3 The points were concentrated in certain places instead of the middle of the triangle. 4 The middle of the triangle remained empty. 8 Table 1 shows that when the number of points in the game increases, some points decrease less in the inner region of the triangle and the points towards the edges are increased. For instance, the findings from the focus group interview with Group 1 are presented below.

Reflections from the activity for relationship between chaos game and Sierpinski triangle
This activity is designed for students to determine the final location of the point using their reasoning skills and to establish a relationship between the steps of the formation of the Sierpinski triangle and the number of dice rolls. In the first part of the activity, 8 groups found a correct relationship between the formation steps of the Sierpinski triangle and the number of dice rolls, while 3 groups did not find a relationship. The relationship obtained by Group 3 is presented in Figure 8. In addition, there are groups that achieve different relationships. For instance, the relationship that Group 5 has achieved is presented in Figure 9. In the second part of the activity, the students were expected to predict the dice that may come according to the last place where the point was found. The prediction of 10 groups were correct, and only the estimates of 1 group were incorrect. The interview with Group 4 which made the correct estimate is presented below: Researcher: How did you determine the numbers on the dice? Group 4: We tried to guess in reverse. Researcher: Can you explain a little more? How? Group 4: Since the end point is near corner B, we thought the dice rolled for the third time could be 3 or 4. Then the other point should be close to point A.
Researcher: Why? Why should it be close to point A. Group 4: Otherwise, point 2 must fall outside the triangle, and our points are always inside the triangle. Researcher: So point 2 cannot be close to C. Group 4: Yes. In the other case where we take the midpoint of the distance, the point falls out. In this case, the 2nd dice may be 1 or 2. Researcher: What would the first dice be then? Group 4: If we continue with the same logic, the first dice should be 5 or 6, so the first point should be close to point C.
From the statements above, it is clear that the students make informed predictions using the available data without measuring

"Reflections from the "Chaos Game and Probability" activity
The purpose of this activity is to enable students to realize that the probability of the numbers in the dice rolled in the Chaos game is equal and to determine in which cases the Sierpinski triangle is formed. All groups performing the activity determined that the formation of the Sierpinski triangle depends on the probability of the numbers on the dice. For instance, the interview with Group 7 is presented below.
Researcher: What relationship did you determine between the probability of the numbers on the dice and the Sierpinski triangle in the game of chaos? Group 7: If we want to create the full triangle, the probability of the numbers coming to each corner must be the same. Otherwise, sometimes a full triangle does not occur. Researcher: What kind of shape is formed? Group 7: At the points where the probability is low, the parts of the triangle with the corners become less faint, indistinct, the other parts become clearer. Researcher: Does a Sierpinski triangle still form? Group 7: Yes, it is formed, but not every corner has the same clarity, some places are more faint. Researcher: Well, have you ever encountered a situation where the Sierpinski triangle does not occur? Group 7: Yes, for example, when we make the probability 1 0 0 or 1 1 0, the Sierpinski triangle does not occur.
Researcher: What kind of shape is formed? Group 7: In 1 0 0 all points are gathered on corner A. Since we always take half of the distance, if every time we throw 1 or 2, the dots are getting closer to corner A and become like a single point. In 1 1 0, a line segment is formed between corner A and B. The logic is the same.
Researcher: So what is the condition for the Sierpinski triangle to form/originate as a result of the Chaos game? Group 7: The probability of all points is the same.
The expressions above show that students realize that the probability of the numbers on the dice must be equal to form the Sierpinski triangle in the Chaos game.

Discussion, Conclusion and Suggestions
In this article, we have described chaos game and provided some tasks for students. Chaos game as a different fractal building method were explained and gave reason for the relationship between Sierpinski triangle and Chaos game. In addition, examples were given from the implementation of the tasks. Students' experiments and reflections on the implementation of tasks were presented. Thus, the adequacy of the task was revealed.
The findings of this study indicated that most students used the observation and hypothesis abilities during the first task. The first activity was prepared for students to create perception about Chaos theory. In this activity, students are expected to realize that there may be a certain order in an event that is seen randomly. In this context, students often used their observation and hypothesis skills during the activity. The students also claimed that the shape that would appear due to the random selection of the starting point at the beginning of the game and the numbers that came in the dice roll could be a random shape. They gave some hypothesis like the final shape should be a filled triangle by dots at the end of the chaos game. However, their later observations caused these hypotheses to change. Their new claim has been that there is no point in the inner region or center of gravity of the triangle. In this context, it is clear that the first activity allows students to use their skills such as hypothesis, observation and inference. In addition, this activity has helped students to monitor the movement of points by measuring in a repetitive process and to make inferences at the end of this process. Play the chaos game on computer revealed that the randomly generated sequence of midpoints increasingly produces a highly structured fractal shape. As the random process was repeated, they noticed that the final shape must be the Sierpinski triangle. In the renewed mathematics curriculum, emphasis is placed on preparing environments and giving examples where students will use their skills such as hypothesis, observation, correlation and inference (MoNE, 2018a; 2018b). In addition, in studies conducted in the literature (Adams & Aslan-Tutak, 2006;Bolte, 2002;Devaney, 2004;Fraboni & Moller, 2008;Karakuş, 2015Karakuş, , 2016Naylor, 1999;Siegrist et al., 2009;Vacc, 1999) it is stated that fractals help students to establish relationships between different subjects of mathematics such as similarity, logarithm, patterns and limit, and enables them to use their skills such as hypothesis, association and inference. The results obtained from this study are similar to the results of the studies in the literature. In the first activity, the Chaos game was played for only three points. This activity can be redesigned with a different number of points. In addition, at the end of the activity, questions can be added to students to form new claims and inferences, such as what kind of forms may occur if the game is played for a different number of points. Also, the game can be replayed for different ratios such as 1/3 or ¼ instead of ½ in the movement of the points. Activities involving these situations and reflections to be taken from students for these activities can be examined in future studies.
National Council of Teachers of Mathematics (NCTM, 2006) mentions the importance of students' recognition, creation and generalization of different patterns in the development of algebraic thinking. The second activity created in this study allows students to discover such patterns. Students made relationship between the number of sub-triangles and the number of the rolls of the die in the second task. Almost every group participating in the study has established correct relationships between the stages of formation of the Sierpinski triangle and the number of dice rolled. In addition, different patterns have emerged, such as the number of dice rolled and the number of triangles formed in the Sierpinski triangle. This shows that the activities designed have the potential to help students find different patterns. In addition, the second activity allows students to use their ability to make predictions and inferences, just as in the first activity. In particular, the section where the last place of the point is given and the place where it was initially asked, enables students to use these skills. In the literature (Adams & Aslan-Tutak, 2006;Fraboni & Moller, 2008;Naylor, 1999;Vacc, 1999;Karakuş, 2015), it is stated that students can reach different generalizations about the area or perimeter of the Sierpinski triangle by using exponent numbers, sequences and limit. In this context, the results obtained from this study coincide with the results of the studies conducted in the literature.
In the last task, students established a relationship between the probabilities of the die and construction of Sierpinski triangle. Findings showed that the tasks enabled students to use their abilities such as hypothesis, observation, mathematical connections and deduction. As a result of the activity, the students concluded that the Sierpinski triangle depends on the probability of numbers on dice which were equal probability. In addition, they had the opportunity to observe what shapes are formed in different probability and in what cases the Sierpinski triangle does not occur. Studies in the literature (Gürbüz, 2006;Işık & Özdemir, 2014;Memnun, 2007) show that the use of concrete materials and worksheets in the teaching of probability topics has a positive effect on students' meaningful learning and academic achievement. The reflections obtained from the activities developed in this study show that activities can help students understand probability topics. In this context, the effects of the tasks on students' meaningful learning and academic success can be examined in future research.
The finding of this study is similar to the literature. It was also determined that the tasks enabled students to make practice on some mathematical topics such as measurement, exponential numbers, probability and patterns. The chaos game not only helps students to build Sierpinski triangle, but also provides a basis for dynamic systems and chaos theory. These tasks can enable students to show interest in or study with these concepts in their future lives. Fractal activities can be found in most NCTM Standards and mathematics curriculums. Thus, fractals can be taught separately or incorporated as examples into traditional lessons.
Chaos game does not only help students create the Sierpinski triangle. At the same time, it provides a foundation for students to realize that regular patterns will occur as a result of random situations, and thus it provides a basis for dynamic systems and chaos theory. In recent years, dynamic systems, fuzzy logic and chaotic structures are among the most frequently discussed topics in the field of mathematics. These prepared activities can allow students to show interest in or work with them in their future lives. Since one of the general objectives of mathematics education is raising future mathematicians (Baki, 2008). As a result of a random situation, the emergence of regular shapes will attract the attention of students at each grade level. Such activities will positively affect students' interests and attitudes towards the mathematics lesson. In the studies conducted in the literature (Ünlü, 2007;Yurtbakan, Aydoğdu-İskenderoğlu ve Sesli, 2016), it is emphasized that the activities and materials to be used in the course have an impact on students' interest in mathematics lesson. In this context, the effects of these designed activities on students' interests and attitudes towards mathematics lesson can be examined in future studies. Students working with the activities prepared in the study have the opportunity to establish relationships between many different mathematical concepts such as probability, number sequences, patterns and measurement. NCTM (2000) emphasizes the importance of working in environments with appropriate activities, materials and examples for students to learn mathematical concepts meaningfully. In this context, the activities prepared have the potential to make different associations between students' mathematical concepts.
In this study, the activities in which students will establish relations with patterns, probability, measurement and number sequences are included. In addition, the importance of integrating information and communication Technologies (ICT) into lessons in mathematics education programs is emphasized (MEB, 2018a;2018b). Considering the teaching of mathematics in our country, it can be said that a traditional approach focused on teachers and the board is adopted (Baki, 2008). In the teaching of mathematics subjects, the rules and features related to the subjects are given by the presentation method, and the subjects are taught with the help of the drawings written on the blackboard. National Council of Teachers of Mathematics (NTCM, 200) emphasizes the importance of using concrete materials, drawings and information and communication technologies in school mathematics. In the activities designed in this study, web site applications prepared for teaching purposes are included. Thus, students were enabled to use technology while establishing these relationships. Thus, the activities designed in this context will contribute to the integration of ICTs into mathematics lessons.