The Challenge: Draw a polygon. Then mark the midpoint of each side. Connect the midpoints of each side to make a new polygon. Repeat. (Don’t sleep on the quadrilaterals here. They do something rather surprising and beautiful!)
Materials Needed: Paper & pencil or online graphing software like Geogebra or Desmos
Math concepts you could explore with this challenge: angles, fractals, functions, geometry, lines, polygons, proportions/ratios, slope symmetry, vertices/intersections.
Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!
1.) Make sense of problems and persevere in solving them. What happens when you do this with a quadrilateral? Will that always be the case? How can you be sure?
8.) Look for and express regularity in repeated reasoning As you iterate (add more layers) what do you notice happening? Is the pattern the same for all polygons? Is there a general description to be made for an n-gon or is it dependent upon the type of polygon you’re working with?
arbitrarilyclose 
The art and math teachers at our school, Whitcomb Middle School in Marlborough, are having our students do this challenge on Friday, 5/8. We had fun making our own examples and their work will be added! Thank you for all your great ideas!
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cooollllllllllll!!!!!!!!!!!!!!!!!
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8uh
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no, it is not cool
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Just five?
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Probably.
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hi
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hi this is awesome and by the way, my name is NOT Anonymous 👍
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I agree with Ms. Perkins…don’t sleep on the quadrilaterals. Something cool happens when you connect the midpoints of the sides on a quadrilateral. It might help to use dynamic geometry tools like Desmos Geometry or Geogebra.
A good follow up investigation would be to try to figure out WHY that works. I used a coordinate geometry approach with a bit of algebraic reasoning. Have fun!
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Cool!
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I can see beautiful triangles!
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this was just too cool! just mind blowing!
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