THE SHOELACE FORMULA IN THE SCOPE OF OUT-OF-SCHOOL MATHEMATICAL EDUCATION - Brydun, Viktoriia; Brydun, Andrii (Physical and Mathematical Education, 2025)
Formulation of the problem. The shoelace formula, also known as Gauss’ formula for calculating the area of a polygon, is important for extracurricular mathematics learning. It helps students understand how to apply mathematical knowledge to real-world problems and demonstrates the practical use of coordinate geometry to calculate the area of any polygon. This approach stimulates the development of spatial thinking, analytical skills, and enables students to solve problems that arise in geography, physics, or architecture. Materials and methods. The study used theoretical and practical methods. Theoretical methods include working with open sources on this topic, analyzing mathematics curricula, and analyzing educational programs for the specialty “Secondary Education. Mathematics”. Practical methods include solving typical problems and exercises on this topic, developing new problems that can be offered to teachers for extracurricular work with students. In addition to the traditional notebook and pencil, the dynamic mathematical software GeoGebra was used to construct polygons. Results. The paper presents the shoelace formula for calculating the area of a polygon with a detailed explanation and proof. An overview of typical problems on this topic is presented and a number of problems are developed that teachers can offer to students within the framework of an optional mathematics course. It is also shown how the shoelace formula can be derived using linear algebra and analytic geometry methods, using determinants and the cross product, and applied to find the areas of curvilinear figures using Green’s theorem. Conclusions. The topics proposed in the work may be useful to mathematics teachers in the context of preparing for specialized Olympiads and conducting electives or math clubs. The relationship between school mathematics and such courses as analytical geometry and mathematical analysis illustrates the need for fundamental basic training for future mathematics teachers.