```The Pythagorean PuzzleEarl Zwicker                   Illinois Institute of Technology                               Biological Chemical and Physical Sci Dept                               IIT Center                               Chicago IL 60616-3793                               (312) 567-3384 Dedication: Thanks to Professor Harald Jensen (1898-1994), PhysicsDepartment, Lake Forest College, who originally worked this idea with thehigh school physics teachers at several summer institutes during the1970s.  This fine example of a phenomenological presentation would notexist if it were not for him.

solid square mentioned in the previous paragraph.  Then let them take it from there with their paper puzzles.  (See Sketch 1.)               The first pair to complete their puzzle should come up and show the rest of us how, using the plastic pieces already on the overhead projector.  (After appropriate applause, etc. award them each a \$100 Grand candy bar, which you have kept out of sight.)      Then say: Thanks!  Now all pairs complete your puzzles!               Everyone complete?  OK!               Now let's see how good you really are. 3.   Can you arrange your puzzle to form two squares of equal area, using all      the pieces? (There is an alternate solution to the first part where there      is an extra rectangle, in which case you omit the phrase 'using all of the      pieces' - but, until they ask, do not tell them that they do not need to use      one of the rectangles.  See Sketch 2.)     Please do so now!  (This will happen quickly for the pieces from sketch 1.)       Then - ask a pair to show their solution using the puzzle pieces already on      the overhead projector. 4. Say: You are now going to prove the Pythagorean theorem.  Can anyone state         what it is?          After brief discussion, project a transparency of the Pythagorean         theorem in words:         For any right triangle, the square of the hypotenuse is equal to the         sum of the squares of the two sides. In other words:        If C is the length of the hypotenuse, and A is the length of its         altitude and B is the length of its base, then C2 = A2 + B2 5. The proof: With the two equal squares projected on the overhead for all to               see, show that they have equal areas by laying them on top of each               other. Make sure that the squares lie on the blank transparency.               Now place them along side each other, and using a felt marker pen,               draw an equal sign between them.      Next, remove two of the four triangles from one square, and one the      rectangles from the other.  Show that the two triangles and the one      rectangle have equal areas (superposition is one easy way).  Since we have      subtracted an equal amount of area from each of the originally equal      squares, the remaining areas must be equal on the left and right sides of      the equal sign.      Again, remove two more triangles from one side of the equal sign, and a      rectangle from the other side. Again, the remaining area on the left     side must equal the area remaining on the right side.     But on one side there will be a small square with the length of its side      equal to the base of a triangle, and a mid-size square with the length of      its side equal to the altitude of the same triangle. On the other side      will be a single largest square with the length of its side equal to the     hypotenuse of the same triangle.  This is easily seen by placing the three      squares on the three appropriate sides of any one of the triangles.Performance Assessment:1. Hand out 2 blank pages and a copy of the rubric to each pair of teachers.   Say: I am going to ask each pair to use your puzzle to prove the Pythagorean         theorem.  Do you want me to take a few minutes to review it?  (They will         say yes.)        OK - there are four steps:  1. left square area = right square area             (Show them again.)     2. subtract equal areas from left and right                                    3. repeat                                     4. remainder areas are equal        2. Now - each pair write up the proof on your page.  Use rough sketches to show         the four steps.  You have about 7 minutes. 3. When finished, exchange your work with a neighboring pair and use the rubric    to score each others work.   References:         If the following graphic does not display or print, contact the author by      letter, telephone or email.  NOTE: the puzzle should be scaled so that the      diagonal square is 3 inches on a side. Sketch 1

In order to draw the puzzle on your own, use 2 sheets of 8.5 x 11 paper, a
pencil, a ruler, a straight edge and a scissors.

Draw a square three inches on a side.  (This is easily done by starting at
one corner of one of the papers and measuring 3 inches down each edge.)
Cut out the square.

Place the square so that its edges lie along the bottom right corner edges
of the second sheet of paper.  Now raise and tilt the square so that its
right bottom corner has moved up the right edge of the page by about 1.5
inches; its left bottom corner should lie at the bottom edge of the page,
about 2.5 inches from the right bottom corner of the page.  The now tilted
bottom of the square will be the hypotenuse of a right triangle, and the
right bottom edges of the page will be the altitude and base of the
triangle.  Use some tape to hold the square in place on the sheet.

Next, draw a horizontal line across the page so that it passes through the
top-most corner of the tilted square.  Then draw a vertical line so that it
passes through the left-most corner of the tilted square.  The square will
now be circumscribed within a larger square formed by the horizontal and
vertical lines drawn on the sheet.  This also leaves the original tilted
square surrounded by four identical triangles; the hypotenuses of the
triangles are the four sides of the tilted square.

For the upper-left triangle, draw a square using one of the triangles
sides as one side of the square.  Draw another square using the other side
of the triangle.

Now draw vertical lines through the vertical sides of the smallest square
(on the left of the triangle).
Then draw horizontal lines through the horizontal sides of the mid-size
square (on the top of the triangle).

You should now have formed three identical rectangles with long sides
vertical (and equal in length to the altitude of the triangles), and short
sides horizontal (and equal in length to the base of the triangles).

Your puzzle is now complete.  Cut it out and play with it.  Enjoy! Sketch 2
(an 'extra' piece for the second part)

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