6-A-3 Pythagorean Puzzles

For the first puzzle, I noticed the length of the white square was ‘c’ and the length of the hypotenuse of the red triangle was ‘c’.  So I rotated each triangle from the previous one to line up side ‘c’ of the triangle to line up with side ‘c’ of the figure. What was left was a white area in the middle that was the exact shape for the blue square to line up with now gaps or overlaps. For the shape on the right, I noticed the veritical side ‘b’ on the white figure, so I lined up the red triangle with the shorter leg ‘b’ with the side ‘b’ in the figure.  I then place the 2nd red triangle in the figure to line up the hypotunuse of the first triangle to form a rectangle.  The length of the bottom of the figure is ‘a+b’, and since I already had a length of ‘a’ lined up on the bottom, I needed to rotate a 3rd red triangle around to line up side ‘b’ with the bottom of the figure. The 4th red triangle, I lined up the hypotenuse with the 3rd to form another rectangle.   What was left was a white square where I knew one side was ‘a-b’, so the blue square fit perfectly.

For the second puzzle, the figure on the left,  I used the same concept as the first figure in the previous puzzle.  I noticed the sides were ‘a+b’, so I used side ‘a’ of one triangle and side ‘b’ of another triangle lined them up with a side of the white square, keeping the right angle of the trangle lined up with the right angle of the white figure.  I rotated the remaining triangles so they lined up with the figure and the right angles lined up with the figure.  What was left was the square shape in the middle, so I rotated the green square it fit.  For the figure on the right, the sides were a+b, so I first put the large blue square in the upper left corner.  Then I took the 2 red triangles and placed them together to form a rectangle and lined it up with an exposed side of the large blue square.  I then took the last 2 red triangles, formed a rectange and lined it up with the other exposed side of the large blue square.  The space that was left  needed a square with the dimentions of ‘b’, which gave the small blue square, with side length ‘b’, a place in the figure.

I didn’t find any of the puzzles to be too difficult because I was looking at the relationships given to me in the pieces and the figures.  If I didn’t have the sides labeled and completing this at random, I would find it more difficult.

I would prefer the hands-on method because I like the idea of physically moving the pieces around, but vitual manipulatives are easier to manipulate since the stay in place better and won’t move around on you.  It really depends on the situation which you have in your classroom.  Depending on what kind of technology you have available would determine which would be better to demonstrate to the class.  If you don’t have many computers available, the hands-on approach would work too.

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One Response to 6-A-3 Pythagorean Puzzles

  1. rwhelchel says:

    Julie,
    I would have saved myself a lot of time and frustration if I had looked at the relationship like you did. I looked at the shapes and tried to just piece it together before I finally got it. I, like you, prefer the hands on method. Moving pieces is more beneficial for students in the classroom. Also, I agree technology can be difficult in some classes depending on the computers and access. It takes our class about five minutes just to get the computers up and running, and our internet doesn’t always work.

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