Finding The Value of π and Circumference of A Circle Formula Through Lid of Cans Like Circle

A branch of mathematics that is taught in school is geometry. There are five levels in geometry learning according to Van Hiele (in yearbook NCTM). The five levels are: level 0 (visualization), level 1 (analysis), level 2 (abstraction), level 3 (deduction), and level 4 (rigor). The level of visualization is often referred to as the recognition rate. Students are familiar with the basic concepts of geometry like square, triangle, rectangle, parallelogram, and others. At analysis level, the students already understand the concept of geometry from informal analysis about the part or the component. At the level of deduction, deductive reasoning students were beginning to develop, but is not well developed. At the level of rigor, students are able to work on the system axioms.

Realistic Mathematics Education is one of the approach that suitable in learning geometry. In this learning process students are given the context of the real world first and then continued with a variety of activities to develop the mindset of students in finding the concepts. Students can develop their understanding of the informal to the formal.

According to the above description, the writer designed the geometry lesson in an elementary school with realistic mathematics education in the sixth grade on the material of a circle to find the value of π and the circumference formula. The learning process is to use the context of lid of cans like circle. This context is taken with the assumption which are often encountered by students in their daily lives. Through this learning activities, the writer wanted to know how the effectiveness of instructional design of realistic mathematics education in the context of lid of can like circle in finding the value of π and the circumference formula.

This learning activity held on Tuesday, November 30, 2012, in the class VIB at Elementary School  98. Students who are involved in the learning process amounts to 33 students consisting 17 men and 16 women.


1.      Preliminary Design

1.1  Curriculum Analysis

In the preliminary design stage, writer analyze the curriculum of grade VI semester 1. The analysis was conducted on the standards of competence, basic competence, the syllabus and materials adapted to Kurikulum Tingkat Satuan Pendidikan (KTSP). Learning activities with realistic mathematics education  will be implemented in the material circle. This material is located on the third competency standards that calculate the area of ​​a simple polygon, area of ​​a circle, and the volume of a triangular prism. While basic competence is calculate the area of ​​a circle.

1.2  Learning Design

Design learning activities are conducted in collaboration with teachers. In these discussions, writer and teacher discusses how learning activities, media are used, the small groups and a conjecture or guess responses of students to the design of learning activities.

Activity 1: calculate the length of diameter of can’s lid like circle

This activity aims to assist students in finding the value of π and circumference of a circle formula. In addition, these activities are also connections between previous knowledge of the parts of a circle with the material to be studied. This is consistent with the characteristics of the intertwining of realistic mathematics education or connections between concepts in mathematics.

Description of activity: Each group received a circular lid, a piece of rope and a circle from carton with the size is same as circular lids. Circle made ​​of carton  given to facilitate students in measuring the diameter of the lid as its center point known. Students are required to determine the length diameter and radius of circular lid.

Activity 2: Measure the length circumference of  can’s lid like circle by using rope

This activity aims to provide an opportunity for students to determine the length of circumference of a circle with an informal manner. This is consistent with the characteristics of realistic mathematics education is informal activities before heading to the formal stage.

Description of activity: Students determine the length of the circumference lid by using a rope. After that the students measure the length of rope used to use  rules.

Activity 3: Find the value of π and circumference of a circle formula

This activity is an informal stage movement that has been done before heading to the formal stage in finding the value of π and circumference of a circle formula.

Description of activity: each group received Student Worksheet (LKS). LKS contains questions relating to the lid held by each group. The question in the LKS are length of the diameter, radius, circumference and the quotient of circumference to the diameter. After that the work of each group is written on the board.

2.      Teaching Experiment

In the learning process, the writer who act as a teacher. Learning activity begins by asking questions to the students about the things around us in the form of a circle as apperception. There are some students who say globe as one of the circular object. Teachers take on the globe and asks the students to observe. Teacher asks students to return to, whether globe including circles or the other. Some students said no and said that the globe including the sphere. In this activity, some students questioned why we should call it the circle whereas we can call round. Responding to a question the teacher replied by saying that in mathematics we call the circle.

Apperception activity followed by asking the parts of a circle and the relationship between the diameter and the radius of which have been studied at the previous meeting. After that the teacher goes on to provide motivation and convey to the student learning goals.

Image 1. Teacher gives apperception about parts circle

After the teacher finished explaining the learning objectives, the teacher began the lesson by asking students which is the circumference of a circle. Students demonstrate the circumference of a circle. Learning activities followed by forming groups of 4 to 5 students. Each group received a circular lid, rope, circle made of carton where the  size is same as circular  lid and student worksheet. Teacher asks students to measure the length of the diameter of the circular lid using a circle made ​​of carton that known the center. In this activity, found that there are some students that measures the diameter of the connecting two points on a circle but not through a central point. To find length of the radius most students do not measure by dividing the diameter.

Image2. Students take measurements of the length of the diameter

The next activity students are given the opportunity to find circumference of a circle by using a long rope given. Students eager for doing this by wrapping the rope to the lid. After that they measure the length of the rope by using the ruler. After the students take measurements, students were asked to fill out worksheets that have been given. In filling LKS given, students have difficulty in finding the quotient between the circumference and diameter. This is because there is a long circumference and diameter of which is a decimal number.

                     (a)                                                                                       (b)

Image 3. (a) Students wrapped the straps on the lid.

(b) students measure the length of rope used

After the students have finished filling worksheets, students were asked to write their answers on the board. Students are required to pay attention to the division of the circumference to the diameter of each group. Each group gets the quotient ranged from 3 to 3.2. The teacher asks the students to find the average value of the quotient. The average distribution yield is 3.11. Teachers giving out guidance to the students that the quotient is a value called the value of π in which the value of π = 22/7 or 3.14. After that, teachers guide students to discover the formula circumference of a circle.

3. Retrospective Analysis

In general, the learning process is going well. Students are actively involved in the learning activities. Students seemed enthusiastic in carrying out the activities required by the teacher.

Before students learn geometry at a more advanced stage, the students should better understand the basics first. In the activity apperception, it is known that students do not understand very well what it is circle. Some students say that the globe also included a circle. In addition to measuring the diameter of the circle in first activity, a number of groups that did not measure the length of the diameter through the center. This is a mistake that will result in students not getting proper diameter. This should have been well understood by students when studying the parts of a circle.

In this activity find the quotient circumference to diameter, many students are having difficulty. This is because there is the length circumference and diameter of which is a decimal number. Elementary School sixth grade students should have understood very well how to do division with decimal numbers.

In realistic mathematics education recognized the iceberg that describe how the students’ understanding of mathematical concepts from the real thing to the formal stage, where students understand abstract mathematical symbols.

image 4. Iceberg of the learning process in finding the value of π and the perimeter of circle formula

C. Conclusion

Learning activities are carried out with realistic mathematical education to increase activity of students and provide opportunities for students to develop patterns of thought. During the learning process of students carry out all activities required teachers well. With this learning process as students become more understanding that mathematics is a human activity. Formulas are usually given by the teacher directly to them actually came from a phenomenon that occurs in the real world and can learn how to find it. Students now know how to get the value of π, and how to find the circumference of a circle formula.

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