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Although mathematics is a fundamental branch of science that can be used in many professions, many high school students believe that mathematics is not important in later life. These students often wonder why they want to study such difficult and, possibly impractical, courses. In this article, we introduce a method to teach the equation of circles, using real-world practical problems. A real-world example involves the use of composite sheets on various convex and concave parts of a building. This teaching method helps students see the importance of mathematics through real-world applications. This article aims to help mathematics teachers prepare appropriate lessons in various areas of mathematics based on the application of mathematics in human life.
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At high school and pre-university level, most math lessons focus on the abstract, which can turn students off and lead them to believe that math is irrelevant for their daily life. From the beginning of the 21st
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Century, educational administrators and teachers have focused on how mathematics can be applied in the real world. According to Vassiliou (2011), since 2007, most European countries have focused on improving students’ mathematical skills to solve real-world problems, rather than theoretical and abstract mathematical concepts . Most students prefer to solve math problems when they can see how their learning can be applied in real life, so considering practical problems suitable for use in the classroom it plays an important role in increasing students’ interest (Vos, 2018).
In educational research, any real-world problem that is solved using mathematics is called a ‘practical problem’ (Gholami & Sathar, 2021; Savizi, 2006). In mathematics lessons, ‘practical problems’ refer to the potential application of this mathematical knowledge in the real world.
In the building facade industry, civil engineers generally use their knowledge and experience to form composite sheets to cover the convex and concave parts of the building. Curved composite sheets are usually attached side by side like a puzzle to cover the facade of a building. However, when using this traditional method, some curved composite sheets cannot be used due to the degree of curvature. Therefore, not only the cost and the amount of work of the engineer are higher, but the quality of the work is affected due to the presence of waves on the surface of the finished work. The convex (concave) part of the building is a circular arch structure. If the engineers have the radius of this circle, they can use advanced engineering machines to bend the composite sheet to the exact required curvature.
The process of finding the radius of a circle is presented as a practical problem to students as a way to learn the concept of the equation of a circle in a more interesting way. In fact, finding the radius of a circle requires the use of certain data about it. Practical problems on the subject of equations of circles are designed as lessons for use in mathematics classes. This course is not limited to a specific field, and mathematics teachers in all countries of the world can use it in their classrooms. Therefore, the purpose of this article is to introduce new methods in teaching mathematics and emphasize the role of practical problems in teaching mathematics.
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A circle is defined as the locus of a point that moves in a plane so that its distance from a fixed point in the plane is always constant. The equation of a circle with center (h, k) and radius r is as follows:
It seems that the use of practical problems that require an understanding of the equation of a circle helps students learn this concept better (Tong et al, 2020). So, after discussing the theory behind the equation of a circle in class, the teacher then introduces this practical problem to the students and asks them to solve it using the equation of a circle and some basic mathematical concepts. If we know the coordinates of at least three points on the circle graph then we can find the equation of the circle through several different solutions.
Figure 1 represents a building with a convex structure; the front part of this part with a composite sheet based on the experience of facing difficulties such as more cost and less quality. In this practical problem, the arc of a circle is given and the engineer needs to have the radius of this circle to bend the composite sheet exactly to the convex part of the building using sophisticated machines. Therefore, the researcher translates these real-world problems into practical problems to provide math lessons to students and asks them to find formulas to solve these practical problems. Therefore, considering this practical problem in the classroom gives students the opportunity to discuss the real-world application of the equation of the circle.
Based on the diagram above, two values h and k are given, so we have the following data about the circle. The point M(0, R) is the center of the circle and three points, N(-k/2, h), E(0, 0) and F(k/2, h) lie on the graph of this circle. We need to find the value for the radius.
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Although there are many ways to solve this practical problem, the researcher discussed two methods to solve it for students to learn some properties of circle equations.
Based on Figure 2, the coordinates of three points N, E and F satisfy the equation of the circle:
So, we can get three equations as follows (in this research, the symbol C(M, R) is used for a circle with center M and radius R).
According to Figure 2, two vertical and horizontal chords of a complete circle intersect at the point (0, h). So, with respect to the above theorem, we have the following equation,
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The use of practical problems and real-world applications of mathematics helps students believe that mathematics is a science that can be used in their everyday lives. This form of solving practical problems is much more interesting for students than solving abstract theoretical problems. Therefore, teachers can increase students’ interest in learning mathematics by considering some practical problems in their lessons.
This study shows the application of the circle equation in the real world about the front side of the convex (concave) part of the building. Based on the mathematical formula in this research, engineers can bend composite pieces such as convex (concave) parts of buildings using sophisticated machines. Therefore, the quality of the building facade has improved significantly. Considering practical problems in this lesson helps students see real-world applications of the equation of a circle. Therefore, learn many new properties about the equation of circles in a more interesting way. This teaching method not only helps students experience the beauty of mathematics, but also shows the importance of mathematics in the real world.
Tong, D H, Loc, N P, Uyen, B P, & Cuong, P ,. (2020). Applying Experiential Learning to Teaching Circle Equations: A Case Study.
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