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I have been planning ahead and it looks like i will have a solid three to four days before the winter break where we wont really be in a unit. We will have just finished a unit and i do not want to start a new one right before such a long break.

So im curious, what are some fun activities or games in the geometry vein that you have done in the past that are appropriate for the high school level?

I'm also open to general critical thinking or reasoning activities, the kids love them and it is still developing their math skills, thank you!

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Have them explore low dimensional topology by cutting up strips of paper: Mobius Strip Activities.

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You might use the amazing "Fold & Cut" Theorem, mentioned in this MESE answer "Secondary Geometry Curriculum Sequencing." Templates available at this web site.

Another idea is to have your students discover the $11$ incongruent "nets" for (unfoldings of) a cube (without knowing in advance that there are $11$):


  CubeNets
      (Figure from this book.)
I hand out pre-grided paper, with cells about 5cm$\times$5cm, and scissors. A class can be fruitfully engaged for quite some time trying to answer the question,

Q. How many distinct (incongruent) nets of the cube are there?

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    $\begingroup$ Oh, I see that the OP also posed the other question to which I linked! $\endgroup$ – Joseph O'Rourke Nov 19 '14 at 17:29
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Another idea, albeit more arithmetic than geometry: Clueless sudoku:


Henle


"Creating Clueless Puzzles." Gerard Butters, Frederick Henle, James Henle, Colleen McGaughey. (PDF download).

There is also an interactive website on which to solve these puzzles.

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lots of great answers! figured i'd contribute for anyone who finds this in the future, I did something similar to @JosephORourkes answer today and it went great! I also have found that Tangrams are something that my students enjoy and they can even be played as a competitive game if they wanted by racing another student to get the correct configuration. Opens up an interesting conversation on the tangram paradox, history, and westernization of the puzzle. Here are some tangram resources and many more printable puzzles can be found via google. I have a few sets of nice plastic ones but in a pinch cutting up some old folders works just as well.

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  • $\begingroup$ I should have thought of Tanagrams! If you can give us any feedback on what worked and what didn't work as well, we'd all learn from your experiences. $\endgroup$ – Joseph O'Rourke Nov 27 '14 at 1:20
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A teacher I once worked with ran a tesselation competition. The students had to create a tesselation similar to those of MC Escher. Points were awarded both for the artistic side (the tesselation pieces looking like something eg an animal, and for use of colour) and for the explanation of how the tesselating shape was created.

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Maybe have the students discover that every triangle can be dissected and rearranged to form a rectangle:


          Fig1.21
          (Figure from this textbook.)
Not easy to discover from scratch. Needs blank paper and lots of scissors.

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This is a supplement to @DavidButlerUofA's answer:


                 
                  (Image from this link.)


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  • $\begingroup$ Thanks! Probably should have thought of that myself. $\endgroup$ – DavidButlerUofA Nov 24 '14 at 1:04

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