THE PYTHAGOREAN PUZZLE by
Earl Zwicker

This lesson was created as a part of the SMART website and is hosted by the Illinois Institute of Technology


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.

Introduction

This is a lesson plan for teachers.

Objectives:

For teachers of teachers (K - Phd): To model the phenomenological approach.
For teachers of students (6 and above): To enable each student to prove the Pythagorean Theorem on his own.
For teachers of students (K - 5): To enable students to identify, and/or define rectangle, square, triangle, and the concept of area (a measure of the amount of surface).

Materials Needed:

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.

Performance Assessment:

References:


Write a letter [c/o BCPS Department:  IIT], telephone me at IIT [1-312-5673384], or email me.  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


(there is an 'extra' piece for the second part)

Who was Pythagoras?


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