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Number of Regions Formed by Connecting Points on the Perimeter of a Circle

Introduction

Learning Mathematics is quite a struggle to many students. Inspite the effort of our educational system to improve students’ performance in the said subject, our country remains and perceived poor when it comes to quality Mathematics education. Edson C. Tandoc Jr. of Inquirer. net specifies a reason on his article published last September 22, 2018 why Filipino students not fit the Asian stereotype. He mentioned there that though Asians are often stereotyped as natural number crunchers, Filipino students seem to be trailing their Asian counterparts in Mathematics and Science. He supported this statement with World Economic Forum’s (WEF) Global Competitiveness Report for 2011-2012 which ranks the Philippines 115th out of 142 countries in perceived quality of Math and Science education, unlike our neighboring countries which belonged to top list like Singapore. This is indeed saddening because Mathematics plays a very important role in our life in general. The study of Braza and Supapo (2014) claimed the shortcomings that can affect the students’ achievement in mathematics especially in Geometry and these are: lack of mastery of the basic concepts and skills, lack of problem solving and critical thinking skills, and students’ lack of connections to the topic they are studying. To resolve these issues, teachers can provide various learning materials that will cater the needs of students. Good thing, Mathematical Investigation can be an avenue to develop students’ problem solving and critical thinking skills as they are forming connection to what they are investigating.

Mathematical Investigation is a sustained exploration of a mathematical situation. Here, the students are expected to pose their own problems after initial exploration of mathematical situation. The exploration of the situation, the formulation of problem and its solution give opportunity for the development of independent mathematical thinking and in engaging in mathematical processes such as organizing and recording data, pattern searching, conjecturing, inferring, justifying and explaining conjectures and generalization.

Learning through mathematical investigation allows students to learn more about mathematics, especially the nature of mathematical activity and thinking. It also makes them realize that learning mathematics involves intuition, systematic exploration, conjecturing and reasoning and not only memorizing and following existing procedures. In this kind of learning, students are given more opportunity to direct their own learning experiences, especially it develops their mathematical habits of mind.

In this paper, a mathematical investigation about the number of regions formed by connecting the points on the perimeter of a circle was shown. There was some prior investigation on how to calculate the number of regions formed by the points on a circle but the author of this paper found out another way of calculating the number of regions formed by doing a mathematical investigation.

Statement of the Problem

This mathematical investigation generally aimed to develop a mathematical formula to calculate the number of regions formed by connecting the points on the perimeter of a circle. Specifically, this investigation aimed to observe some patterns that will help the investigator develop a conjecture.

Scope and Delimitation

The focus of this mathematical investigation was to develop a mathematical formula to calculate the number of regions formed by connecting the points on the perimeter of a circle. The investigator drawn five (5) circles, put one (1) point on the first circle, two (2) points on the second d circle, three (3) points on the third circle, four (4) points on the fourth circle and five (5) points on the fifth circle respectively. Then, the investigator observed some patterns and form a conjecture. The investigator illustrated a circle with at most ten (10) points on it and list some observations.

Significance of the Study

This mathematical investigation was designed with the expectation that the outcome may be essential in uplifting the quality of education in the country by developing and enhancing the students’ problem solving and critical thinking skills, strengthening their connections to what they are studying and mastering some concepts in Math through the process of Mathematical Investigation. In particular, the findings of this study will be significant to the students, teachers, administrators, education sectors and other stakeholders.

There are several interesting concepts and problems that arise from looking at a circle with points on its perimeter. There are several ideas to explore relating to this, but this mathematical investigation specifically takes a look at three (3) things:

First, given a circle, what is the number of lines or chords determined by points on its perimeter? Chord refers to the segment joining any two points on a circle.

Second, given a circle, what is the number of intersections formed by the chords or lines in a circle? The intersection is a point wherein two or more chords/lines meet.

Third, given a circle with lines or chords that are formed by connecting the points on its perimeter, what is the number of regions that can be formed? Below is a visual representation of what is meant by the term region above.

number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle

To present the investigation, ten (10) figures were shown below. Each figure shows theu number of points on a circle, (b) number of lines formed by the points on a circle (c) number of interior intersections of points and (d) number of regions formed. The number of points, lines, intersections, and regions were patiently counted by the investigator. Then, these data were observed to form a conjecture.

number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle
number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle
number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle
number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle

Observations

1. A circle itself determines one (1) region.

2. As the number of points on a circle increases, the number of lines/chords also increases.

3. Each time the number of points on a circle increases by 1, the number of lines/chords increases and is equal to “previous number of lines/chords + (n − 1)”. A table of increase is shown below.

number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle

4. The number of lines/chords is equal to the number of ways 2 points can be chosen from points or is represented by ∁(n, 2). (Note: r = 2 because 2 points determine a line.)

number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle
number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle

5. The number of interior intersections is equal to the number of quadrilaterals that can be formed from points since each quadrilateral produces just one intersection where the diagonals of the quadrilateral intersect. Thus, the number of interior intersections is represented by ∁(n, 4). (Note: r = 4 since 4 points determine a quadrilateral)

number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle

6. As the number of points, lines, and interior intersection increases, the number of regions formed also increases.

7. The number of region is equal to the sum of 1, number of lines/chords, and number of interior intersection of a circle. One (1) is a constant since a circle determines one region as stated in the first observation. Below is a table for the computation of the number of regions.

number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle

Conjectures

number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle

Conclusions

number-of-regions-formed-by-connecting-points-on-the-perimeter-of-a-circle