La sezione aurea come linguaggio matematico della bellezza

AUTHOR: Annalisa Papasidero CONTEXT: Laboratory, non-mathematical, mathematical, aimed at an audience of high school seniors. TOOLS: Graph paper, compass, pencil, ruler. OBJECTIVES: Construct and recognize the golden rectangle. Understand the correlation between the golden ratio and the Fibonacci sequence. CONCEPTUAL CORE Geometry Arithmetic and Algebra Relations and Functions Argumentation and Conjecture Solving and posing problems CONCEPTUAL CORE: Approximations, plane figures, sequences, proportions. ACTIVITY DESCRIPTION: Phase 1: Introduce the definition of the golden ratio. Construct a golden rectangle using a straightedge and compass. Step 2: The resulting golden rectangle is considered, and by repeatedly folding the paper, a series of increasingly smaller squares are obtained within the sheet. Note that the lengths of the sides of the squares constitute some terms of the Fibonacci sequence. Step 3: Circular arcs are drawn within the squares. Note how the union of lines forms a spiral, a shape often found in nature. Step 4: Photos of some examples of golden ratios in art and nature are shown. REFERENCES TO NATIONAL GUIDELINES: Learning objectives at the end of the second two-year period of high school: The study of geometry in the second two-year period will consider the extension of some of the themes of plane geometry to space, also with the aim of developing geometric intuition. The topic of approximate calculus will also be addressed, both theoretically and through the use of calculation tools. Students will acquire knowledge of simple examples of numerical sequences, including those defined by recurrence, and will be able to handle situations involving arithmetic and geometric progressions. SKILL DEVELOPMENT GOALS AT THE END OF SCIENTIFIC HIGH SCHOOL: At the end of the scientific high school program, students will understand the basic concepts and methods of mathematics, both within the discipline itself and relevant to the description and prediction of phenomena, particularly in the physical world. They will be able to place the various mathematical theories studied within the historical context in which they developed and will understand their conceptual significance. Students will have acquired a historical-critical understanding of the relationships between the main themes of mathematical thought and the philosophical, scientific, and technological contexts.