Teaching and Learning Greatest Common Factor (GCF) and Least Common Multiple (LCM) at Grade V Using RME Approach

 A.    INTRODUCTION

This observation report explains about the process of teaching and learning Greatest Common Factor (GCF) and Least Common Multiple (LCM) at SD Negeri 98 Palembang, South Sumatera, Indonesia by using Realistic Mathematics Education Approach. This activity involved 33 pupils in two days.

In the first meeting, I was a teacher in the classroom. The teaching and learning activities ran about 60 minutes which was helped by Mrs. Maryani and Novita Sari for documenting and guiding the students to follow the instruction. In this meeting, we introduced the concept of Greatest Common Factor (GCF) to the students by giving the realistic problem.

In the Second meeting, in the second day, Novita Sari acted as a teacher and helped by Mrs. Maryani and me to guide the students and document the teaching and learning process. In this chance, we introduced the concept of Least Common Multiple (LCM) by using the realistic problem as we did in the first meeting.

B.     GOAL OF RESEARCH

The goal of this research is to make students understand the concept of Greatest Common Factor and Least Common Multiple by using Realistic Mathematics Education Approach.

C.    RESEARCH FINDING

1.      Discussion Result

This research conducted in SD Negeri 98 Palembang, a public primary school located at K.H.A. Azhari street, 14 Ulu, Palembang on the 13th and 14th of September, 2011. It is one of the public primary schools in Palembang that has been applying PMRI in the classroom. Before my friend Novita sari and I, went to this school for conducting our research, we had had discussion with Mrs. Maryani as a teacher at grade v. From the discussion, we got the information that the students have already learned about prime factorization. So, based upon this discussion, we decided to design our research in the topics of GCF and LCM.

2.      Research Activities

Greatest Common Factor (GCF)

In the first meeting, we tried to explore about how far the students understanding about the concept of factorization by using leaves as a media of instruction. We divided the students into small groups consist of 5 and 6 members. We guided the students to divide the leaves either equally or not into small groups, by using 20 leaves.

Students try to divide the 20 leaves equally and not equally.

Figure 1. Students acitivity

Students try to divide the 20 leaves equally and not equally.

Figure 2. Students try to divide the 20 leaves equally and not equally.

Most of students are quite easily to divide the 20 leaves either equally or not because they have already known the concept of factorization. But, there are still some of them that get confuse at the first time. After giving an explanation from Mrs. Maryani and Novita Sari, they finally can do it. In that time, we not only asked the students to divide 20 leaves, but also to write the result in their books and make a list. In the end of activity the students realized that the leaves can be divided into small groups equally without remainder and sometimes cannot be equally with remainders.

Student’s list

Figure 3. Student’s list

After learning about factorization, we introduced the concept of Greatest Common Factor (hereafter GCF) by using realistic problem. We gave the problem by telling the story that we have 20 chocolate candies and 15 fruit candies to give to the students equally. The question is how many students that we have to give in order to give them equally. After giving several minutes to think, we invited one of the students, M. Junaidi, as a volunteer to divide and give all the candies to his friends. When he came to in front of the class, he thought several minutes about the problem. After that, I asked him to decide how many friends of him that he is going to invite and finally he comes to the answer 5 students. He invited five of his friends to come to the in front of the class and give them the candies. After thinking for a while, he become easily divide and give the 20 chocolate and 15 fruit candies to his five friends, each of them get 4 chocolate candies and 3 fruit candies. Realizing that Junaidi is easily solved the problem; we try to ask him why he directly invited five of his friends. He just answers that 20 and 15 is can be divided by 5 without remainders. Furthermore, we try to explore more about the other students’ understanding about the problem by asking them about the reason why when Junaidi invite five of his friends to get the candies, there is no candy left. Some of them have the same answer as Junaidi said. But, there are also some of them said that 5 is the factor of 20 and 15. In the end, Mrs. Maryani explained to them that 5 is the Greatest Common Factor (GCF) of 20 and 15. So, that is why all of the candies can be divided equally.

Junaidi gives all of the candies to his friends

Figure 4. Junaidi gives all of the candies to his friends

Least Common Multiple (LCM)

In the second meeting, my friend Novita sari leads the classroom as a teacher. In this chance, she will explain the concept of Least Common Multiple (hereafter LCM) by using games and realistic problem in the student worksheet. She starts the teaching and learning activities by giving game namely “tepuk tangan gembira” or “happy clap” in English.

Before starting the game, the teacher checks the students understanding about the concept of multiples. She asks about what are the multiples of two. In fact, several students pronounced the answers such as 4, 6, 8, 10, 12, etc. It is clearly described that the students have already known the concept of multiple.

The teacher checks the students understanding about the concept of multiples

Figure 5. The teacher checks the students understanding about the concept of multiples

Later on, the teacher explains about the roles of the game that is the teacher, Novita Sari, will count 20 first natural numbers and every time she come to 2 and its multiples, the students have to clap together. At the first time, some students get confuse and they still clap their hands even if the teacher did not say the multiples of 2. So, based upon this condition, the teacher, Mrs. Maryani and I explain to the students the role once again. After that, the teacher said the numbers again consecutively from one to twenty and that time most of the students successfully do the game.

The teacher explains the role of the game

Figure 6. The teacher explains the role of the game

Then, the teacher decided to increase the difficult of the game by divide the students into two big groups. She also changes the roles that the first group claps their hands when the teacher says 2 and its multiples. The second group claps their hands when the teacher says 3 and its multiples. After explaining the roles to the students, the teacher starts the game and count the natural numbers until 20. But, a little bit crowded claps come to the classroom. Several students seems not understand the role well or perhaps forget the multiple of 2 and 3. Most of them clap their hands when numbers said by the teacher come to even or odd. Realizing that fact, the teacher, Mrs. Maryani and I explain again about the roles of the game that they only clap their hands when the numbers are multiples of 2 or 3 not even or odd numbers. Finally, after explaining this role again, most of them successfully do the game.

The students in the first group clap their hands together

Figure 7. The students in the first group clap their hands together

Furthermore, the teacher asks when they clap their hands together. Some of them answer this question that they are clapping their hands together when the numbers are 6, 12 and 18. Following that answer, the teacher asks them to write down the answer in the whiteboard.

Aisyah Utami writes her answer in the whiteboard

Figure 8. Aisyah Utami writes her answer in the whiteboard

Later on, the teacher asks them again why they clap their hands together in those numbers, but not in the others. In the beginning, no one can explain the reason. After several minutes waiting the answer from the students, Mrs. Maryani gives some clues to them and the students finally know the reason. Unfortunately, she actually gives the answer instead of the clues. She said you did not clap your hands together in the number 10 because 10 is not multiple of thrrr…. and the students just completed the answer by saying three.

This fact makes the teacher, explain more about common multiple of two numbers. She makes an example of Sriwijaya Empire and Dutch colonizer. She asks the students what if the Sriwijaya Empire affiliates with the Dutch as a colonizer. What is the name of this condition. One of students said in Indonesia “bersatu” and the other said “persekutuan”. So, based upon that story, the students know what the meaning of “persekutuan” or common in English.

Following this game, the teacher gives the realistic problem to the students in order to more understand about the concept of LCM. The problem is written in the worksheet.

The realistic problem

Figure 9. The realistic problem

Before solving the problem, the students divided into small group as we did in the first meeting. They are doing and trying to solve the problem together with their friends.

Students activities to solve the problem

Figure 10. Students activities to solve the problem

While students work to solve the problem, Mrs. Maryani, Novita Sari, and I are going around to see what they do and sometimes help them to think about the solution by giving a clue, especially to the lower achiever students.

The teacher comes to the help the students

Figure 11. The teacher comes to the help the students

After several minutes, the teacher asks them whether they have done or not. Once they said “yes, we done”, the teacher asks anyone who wants to present their work. But, unfortunately, there is no one to do that. So, the teacher asks them to write their work in the whiteboard.

The students write her group’s answer in the whiteboard

Figure 12. The students write her group’s answer in the whiteboard

Even though, there are no one can present their answer, there are several ways found by the students to solve the problem. One of them is using the calendar in the worksheet and circles the multiple of 3 and 5 there, as the fifth group did. The other is listing the multiples of 3 and 5 like the first group did.

In the end of this meeting, the teacher explains the connection between the answer and the concept of LCM because no one student wants to do it, even though they have already known. Because the time for teaching mathematics in that class is over, the teacher closes that meeting and say good bye to the students.


ICEBERG

Iceberg in teaching and learning GCF

Figure 13. Iceberg in teaching and learning GCF

Iceberg in teaching and learning LCM

Figure 14. Iceberg in teaching and learning LCM

CONCLUSION

In conclusion, we have already known that teaching and learning Greatest Common Factor and Least Common Multiple are really interesting if we use Realistic Mathematics Education Approach. Most of the students in the classroom are really active and they feel happy learn mathematics in this way. Besides this fact, there are still some problems that we are going to solve in the next design research or instructional design such as the way we inform the students about the problem or the way we manage the classroom. Because, according to our experience in this research, the students sometimes get confuse with our explanation. Other than that, we also have to learn more about RME approach and the other concepts relating to this such as Context in teaching mathematics and Hypothetical Learning Trajectory.

References:

De Lange Jzn, J. 1987. Mathematics, Insight and Meaning. Utrecht: OW & OC, Utrecht University.

Freudenthal, H. 1973. Mathematics as an Educational Task. Dordrecht: Reidel Publishing Company.

Freudenthal, H. 1978. Weeding and Sowing, A Preface to a Science and Mathematics Education. Dordrecht: Reidel Publishing Company.

Freudenthal, H. 1991. Revisiting Mathematics education, China Lectures. Dordrecht: Kluwer Academic Publishers.

Freudenthal, H. 1999. Didactical Phenomenology of Mathematical Structure. Dordrecht: Kluwer Academic Publishers.

Gravemeijer, K. 1994. Developing Realistic Mathematics Education. CD-B Press, Center for Science and Mathematics Education, Freudenthal Institute, Utrecht University.

Prahmana, R.C.I. 2010. Batu, Permen, dan Berbagi yang Adil. Majalah PMRI Vol. VIII. No. 3. Institut Pengembang Pendidikan Matematika Realistik Indonesia.

Sembiring, R., Hoogland, K. and Dolk, M. 2010. A Decade of PMRI in Indonesia. The Netherlands: APS International.

Streefland, L. 1991. Fractions in Realistic Mathematics Education. Dordrecht: Kluwer Academic Publishers.

Zulkardi. 2002. Developing a Learning Environment on Realistic Mathematics Education for Indonesian Student Teachers. Thesis. University of Twente. Enschede: PrintPartners Ipskamp.

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