Meaning of Exponents via Geometric Visualization

① This example demonstrates, through visual geometry, how exponents (powers) can be interpreted as repeated multiplication or scaling effects, with emphasis on how a piecewise (different-case) construction changes the outcome. ② Key ideas include the meaning of exponents, the link between powers and geometric quantities (such as lengths/areas/scales), and the role of piecewise structure or conditional definitions in building the model. ③ Construction highlights: it uses the GeoGebra commands Polygon and Sequence to create a polygonal framework and to generate a point sequence determined by exponent-related rules; it relies on geometric objects such as a, b, c, A, e, poly1, f, g, h, i, and list1 to define vertices/edges and the reusable structure; labels (text1, text2, text3, etc.) annotate the exponent meaning and the correspondence; interactive controls like button1 and button2 support switching between cases to observe how different powers affect the diagram. ④ Suitable for middle to high school mathematics lessons and learning activities on exponents and basic modeling, helping students connect abstract exponent rules to an intuitive geometric interpretation.