THE
PYTHAGOREAN PUZZLE
by Earl Zwicker
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.
- If you want other teachers to understand the
phenomenological
approach to
teaching/learning, this is a good way to begin.
- If you want your students to understand the Pythagorean
Theorem, and even
prove it, then follow the ideas below.
- If you would like a quick review of the theorem and its proofs,
try this.
More!
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:
- 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.
- 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.
- 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.), then pass out a sandwich bag of
puzzle pieces
to each pair. Tell them they have just 5 minutes to solve
and win a $100
Grand prize.
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.
- 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.
- 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}
- 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:
- 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:
- left square area = right square area (Show
them again.)
- subtract equal areas from left and right
- repeat
- remainder areas are equal
- Now - each pair write up the proof on your page. Use rough
sketches to
show the four steps. You have
about 7 minutes.
- 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)
Who was Pythagoras?
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© Earl Zwicker
Last Updated: Thursday, August 19, 2004