SHARPEST ISOSCELES TRIANGLES ON ODD-POLYGONS<\/strong><\/p>\n Excel file: 6. Perfect Squares<\/a><\/p>\n Paper: 6._Alternative Visions of Perfect Squares<\/a><\/p>\n This is a fun exercise that can be used as soon as students know about multiplication. This exercise shows a very slick way of counting sharpest isosceles triangles on odd regular polygons by showing the connection between the triangular image inscribed in a polygon to dots organized into a perfect square array. The Excel file is a self-contained teaching tool which relies on a series of click-boxes that build out the model via a series of discussion points or questions on the Sharpest Triangles <\/strong>sheet.<\/p>\n This is not the first piece I wrote with Don Chakerian but it is the one that can be explained all the way down to third grade. (We initially worked on counting sharpest isosceles triangles on even regular polygons because those images emerged from File 4. Only later did we discover that the odd cases had easier answers although they required a different way of creating the images. We will look at these images using File 8.) The most comprehensive discussion of counting exercises that we have is in the paper:<\/p>\n “Up\u00a0 the Hill and Down Again,”<\/em> The College Mathematics Journal<\/strong>, Vol. 54, No. 4, September 2023, pp. 253-265 plus online supplements.<\/p>\n Unfortunately, CMJ<\/strong> is not open-source.<\/p>\n This is an alternative way to present this material based on Dan Mayer’s Three Act Math<\/em><\/a>. The student version is in this Excel file<\/a>, and instructor notes are here<\/a>. James Marks ’23 wrote the Three Act<\/em> prompts in the student version as well as the instructor notes.<\/p>\n The explainers below are a bit more linear than in the string art portion of PwP,<\/strong> but they are shown in small, bite-sized pieces rather than as a larger more academic style paper posted above (which covers the first 5 sections and triangular numbers below).<\/p>\n P3. Sharpest Odd Isosceles Triangles<\/strong><\/p>\n Focus on Overlapping Triangles<\/p>\n Focus on Smallest Pieces (Triangles and Trapezoids)<\/p>\n An Additional Pattern and Concluding Thoughts<\/p>\n P8. Squares and Cubes.<\/strong> This is a bit of a detour from counting triangles but because it deals with perfect squares it is sitting here rather than at the end of the counting analysis. Two additional counting patterns are analyzed here, one for squares and one for cubes.<\/p>\n\n
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