Return to Physics IndexThe Pythagorean Puzzle

Earl 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), 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 musttellwhat 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 theywantyou to do, but rather only and literally what theytellyou 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 asingle, solidsquare.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 thesingle,solid squarementioned 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 C^{2}= A^{2}+ B^{2}

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 1In 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)