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# Geoboard Problems for Teachers

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1 thought on “Geoboard Problems for Teachers”

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Hi,I really enjoyed thinking about some of the subproblems here. You'll have to imagine my yellow-pad full of tangents.Among other ones that might be of interest: Why is there a triangle of area 15 at all as opposed to some irrational value, do all the possible triangles on the board have interesting areas? Yes: they're either integers or multiples of 1/2. This flows out of boxing the triangles in and calculating the area via subtraction of the right triangles on the edges (which all are also integral or multiples of 1/2).Find the areas via integer factorizations:For instance on an 7 x 6 box that encloses the triangle in your picture you get the equation for the area of any triangle of this basic form as 7 x 6 – 1/2xy – 1/2(7-x)6 – 1/2(6-y)7 = 21 – 1/2(7 – x)(6 – y)If you set it it any particular desired value like 15 then you just have to check the factorizations:So for 15 you get 12 = (7 – x)(6 – y) and you need to check (1,12)(2,6),(3,4),(4,3),(6,2) and (12,1)(3,4) corresponds to your picture. All the other possible ones flow out in the same way and then you obviously rotate or reflect them.It was then fun to think about the super-obtuse triangles that don't fit this model i.e. two vertices must be at opposite corners with the 3rd on the inside of the box.Finally, I thought checking all the possible areas for a given box was fun too. It relates mostly to the number of factorizations for each size biased by cutoffs where the factorizations are not possible. Thanks

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