The Pythagorean Puzzle

Earl Zwicker                   Illinois Institute of Technology
1809 Heather Court             Biological Chemical and Physical Sci Dept
Northbrook IL 60062 5869       IIT Center
(847) 272-6395                 Chicago IL 60616-3793
zwicker@charlie.iit.edu        (312) 567-3384
 

Dedication: Thanks to Professor Harald Jensen (1898-1994), Physics
Department, Lake Forest College, who originally worked this idea with the
high school physics teachers at several summer institutes during the
1970s.  This fine example of a phenomenological presentation would not
exist if it were not for him.

Objectives:

For teachers (K - Phd): To model the phenomenological approach.

For students (grades 6 and above): To enable each student to prove the 
                                   Pythagorean Theorem on his own.

For students (grades K - 5): To enable students to identify, name and/or define 
                             rectangle, square, triangle, and the concept of
                             area (a measure of the amount of surface).

Materials Needed:

     Pieces for Pythagorean puzzle sawed from colored, transparent plastic 
     sheet (all the same thickness - about 1/8 inch - pieces of 4 different 
     colors).  Optionally - several $100 Grand candy bars.

     Sandwich bags, one for each person, each bag containing a set of the 
     puzzle pieces cut from brightly colored paper.  (Use a machine to copy the 
     puzzle onto sheets of yellow paper and cut out.)

     Performance assessment rubric.

Strategy:

For teachers:  Ask: "Who has seen this before; anyone?  Raise your hands."
If any hands are raised, then announce - "I need your help!  If you have seen 
or done this before, please do not give it away to those who have not.  Please 
don't spoil their fun."

     Next, ask people to form pairs or partners by holding their hands up 
together.  Tell them to remember who their partner is.

1. Have the overhead projector prepared ahead of time by placing a blank 
   transparency centered on its projection area.  Then begin by placing the 
   pieces of the puzzle on the overhead and viewing the image on a screen
   so all can see and participate.

   Challenge teachers to tell you how to assemble the pieces into a solid 
   rectangle using all the pieces - they must tell what to do; cannot show.

   NOTE: Invariably, teachers - or almost anyone for that matter - will find it 
difficult to tell you what to do.  e.g. They might say, "Move the piece on top 
next to the gray one."  And you will move the piece, but not place them in 
contact; or you will move the wrong piece, etc.  You will not automatically do 
what they want you to do, but rather only and literally what they 
tell you to do.  They will laugh to see how "stupid" you seem to be, but 
they will see that you are doing only what they told you to do. 

     After 5 minutes or so, somebody might use the word "triangle" or "square" 
or "rectangle" to describe the piece they wish you to move.  As soon as one of 
these words is used, repeat the word several times, (e.g. "triangle") and ask a 
volunteer to define the word.  Ask them to name the other pieces and get their 
definition for each piece until everyone agrees and understands correctly the 
names of the pieces.  For triangles, make sure everyone agrees to the meanings 
of "hypotenuse", "altitude", and "base". 

     This brings out the need for a common vocabulary, and the need to be able 
to express one's thoughts with precision.  If, then, someone asks you to move a 
green triangle adjacent to the large, orange square, you will do so, but again, 
the result is not what the person intended for you to do. You then might ask, 
"Do you mean that you want me to move a green triangle so that its hypotenuse is 
in continuous contact with an entire side of the large orange square?"  If they 
express agreement, then do it.  Then see if others can express their thoughts 
with precision by telling you what to do next.  But once the point is made, do 
not belabor it; go on to the next step. 

2.  Solve the puzzle. 
    
    See if they can direct you to the point where you have placed each of the 
four identical triangles so that each has its hypotenuse congruent with one of 
the four sides of the largest square, thus forming a single, solid 
square.  Once this is done, solving the puzzle will proceed rapidly. But 
if more than 10 - 13 minutes have passed (aside from the digressions into the 
need for vocabulary, etc.) and they still haven't solved the puzzle (which is  
usually what happens), then pass out a sandwich bag of puzzle pieces to 
each pair.  

     Then challenge each pair to complete the puzzle to form a solid rectangle 
using their pieces.  NOTE: If no pair succeeds within 5 minutes, then give a 
hint: Using the pieces on the overhead, show them how to form the single, 
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)