Basic geometric-trigonometric equalities and problem-solving: Linear and angular elements
This paper presents some equalities that we consider useful for the resolution of general high school level geometry problems. The presented equalities involve the use of linear elements and angular elements at the same time. The objective of this work is to monitor, on a small scale, the skills and knowledge acquired by students in the eleventh and twelfth school year (L11 & L12) in an Italian region named Apulia. This goal has been achieved by mean of different exercises where we asked 27 students:
- recognise the structure of proposed equality;
- resolve expressions with equalities.
Results are reported and discussed.
Anonymous, (1904). Relations entre les éléments d’un triangle. Vuibert et Nony, Éditeurs, Paris.
Altshiller-Court N., (2007). College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle, Dover Publications.
Bottema O., Djordjevic R.Z., Janic R.R., Mitrinovic D.S., Vasic P.M., (1969). Geometric Inequalities, Wolters-Noordho Publishing Groningen.
Duckworth A., (2016). Grit. The power of passion and perseverance, Scribner.
Engel A., (1998). Problem Solving Strategies, Springer Verlang.
Hang K.H., Wang H., (2017). Solving Problems in Geometry: Insights and Strategies, World Scientific Pub Co Inc.
Hobson E.W., (2005). A Treatise on Plane and Advanced Trigonometry, Dover Publications, Inc. Mineola, New York.
Larson L.C., (1983). Problem-Solving Through Problems, Springer-Verlag.
Ligouras P., (2017a). Basic Geometric Equalities and Problem-Solving: Linear Elements, Experiences of Teaching with Mathematics, Sciences and Technology — ISSN 2421-7247, vol. 3, n. 1, 475-487.
Ligouras P., (2017b). Basic Geometric Equalities and Problem-Solving: Angular Elements, Experiences of Teaching with Mathematics, Sciences and Technology — ISSN 2421-7247, vol. 3, n. 2, 515-526.
Ligouras P., (2010). 1691 Algebraic Inequalities: Problem Solving – Old and New Problems for the Mathematical Olympiads, AGA editrice.
Ligouras P., (2008). Geometrical Olympiad 2008, AGA editrice, ISBN: 88-95089-11-9 (Italian).
Prasolov V.V., Tikhomirov V.M., (2001). Geometry, American Mathematical Society.
Shariguin I., (1989). Problemas de Geometría Planimetría. Editorial Mir.
Todhunter I., Problems and Solutions in Plane Trigonometry (LaTeX Edition): For the use of Colleges and Schools, Ancient Science Publishers, 2016.
Vygotsky L.S., (2012). Thought and Language, 2nd Edition, The MIT Press.
Zeitz P., (2006). The Art and Craft of Problem Solving, Wiley International Student edition.