Pick's formula is a way to find the area of a geoboard polygon by counting interior pegs and boundary pegs. Students can discover the formula by doing some experimenting under teacher guidance (see Geometry Labs 8.6 or Algebra: Themes, Tools, Concepts 4.12.) I have used this in the classroom for decades, because it is such a beautiful and surprising result, which provides a wonderful connection between algebra and geometry, and because it is both accessible and challenging in just the right way. But it's only three years ago, at a meeting of Escape from the Textbook! that I started to understand the underlying math.
Read all about this on a new page, for teachers, on my Web site: Proving Pick's Formula. There, I provide an illustrated step-by-step proof of why the formula will always work. I leave some of the algebra for you to do, so get a pencil and paper before clicking.
Many thanks to Kim Seashore, Avery Pickford, and Dan Bennett for their help with this!
For more lattice / geoboard math on this blog, click here.
I have retired from the classroom after 42 years as a math teacher in K-12 — from counting to calculus. I now work with teachers and schools, and I continue to develop curriculum.
I share instructional materials on my Math Education Page, and my views on math education on this blog. Also on this blog, announcements about: new pages, updates, etc. on the site; my publications; my appearances at conferences; and my other activities in math education.
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All the comments between August 2013 and June 2016 got deleted at once by Google when I disconnected the comments from Google+. I had to disconnect because otherwise it was impossible to remove spam. Apologies to all the commenters.