**SHARPEST ISOSCELES TRIANGLES ON ODD-POLYGONS**

Excel file: 6. Perfect Squares

Paper: 6._Alternative Visions of Perfect Squares

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 **sheet.

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:

*“Up the Hill and Down Again,”* **The College Mathematics Journal**, Vol. 54, No. 4, September 2023, pp. 253-265 plus online supplements.

Unfortunately, **CMJ** is not open-source.

This is an alternative way to present this material based on Dan Mayer’s *Three Act Math*. The student version is in this Excel file, and instructor notes are here. James Marks ’23 wrote the *Three Act* prompts in the student version as well as the instructor notes.

The explainers below are a bit more linear than in the string art portion of** PwP,** 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).

- Alternative Visions of Perfect Squares, An Introduction
- Creating an Image based on Sharpest Isosceles Triangles on Odd Polygonal Vertices

Focus on Overlapping Triangles

- Counting Triangles
- Looking at Squares along Diagonals
- A One Sentence Answer. Look at Apex Counts Zig-Zag from Side to Side
- The
*Counting Exercise Handout i*s a useful teaching companion to this part.

Focus on Smallest Pieces

- Nonoverlapping Polygons
- Looking at Squares by Gnomons
- Another One Sentence Answer. Add Row Counts Alternating Top and Bottom to find T&T

Two Additional Patterns and Concluding Thoughts