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Activity I: Proving the Pythagorean Theorem (Grade Levels: 6-9)
About Math Concepts |
Proving the Pythagorean Theorem |
Magic Squares and Stars |
The Tower of Hanoi |
More Math Concepts
Standards:
Standard 7: Reasoning and Proof
Objectives:
The Pythagorean theorem can be proven using several different basic figures. This activity introduces student to two such figures with a brief explanation of how to go about the proof. The activity will demonstrate alternate solutions to the problems as well as provide a glimpse into the way early mathematicians reasoned about mathematics.
At the end of this activity the students should be able to:
- understand that there are many ways to approach a problem
- not to be completely reliant on a given drawing
- recognize that not all geometric proofs are two column deductive proofs so familiar in traditional textbooks.
Activity 1
This activity is a high school level activity that can be adapted for middle school and upper elementary students by simply having the students determine as many as possible Pythagorean triples.
Proving the Pythagorean Theorem
Pythagoras
of Samos, c.560–480 BC, was a Greek philosopher and
religious leader who was responsible for important developments
in the history of mathematics, astronomy, and the theory
of music. He migrated to Croton where he founded a philosophical
and religious school that attracted many followers.
Because no reliable contemporary records survive, and
because the school practiced both secrecy and communalism,
the contributions of Pythagoras himself and those of
his followers cannot be distinguished. The most important
discovery of this school was the fact that the diagonal
of a square is not a rational multiple of its side (that
is, the diagonal of a square is not a number that can
be expressed as the ratio of two whole numbers.). In
essence, this showed the existence of irrational numbers.
This discovery disturbed Greek mathematicians and the
Pythagoreans themselves, who believed that whole numbers
and their ratios could account for geometrical properties.
Pythagoreans believed that all relations could be reduced
to number relations ("all things are numbers").
The
Pythagoreans knew, as did the Egyptians before them,
that any triangle whose sides were in the ratio 3:4:5
was a right-angled triangle. The so-called Pythagorean
theorem, that the square of the hypotenuse of a right
triangle is equal to the sum of the squares of the other
two sides, may have been known in Babylonia, where Pythagoras
traveled in his youth. The Pythagoreans, however, are
usually credited with the first proof of this theorem.
Much
of the Pythagorean doctrine that has survived consists
of numerology and number mysticism, and the influence
of the belief that the world can be understood through
mathematics. That belief was extremely important to
the development of science and mathematics.
Proving
the Pythagorean Theorem
The following figure
is the typical figure used to prove the Pythagorean theorem that the
square of the hypotenuse of a right triangle is equal to the sum of the
squares of the other two sides. c2 = a2
+ b2
1. If the right
triangle ACB were isosceles, the figure would appear as follows.
Show how with the
addition of three lines the proof of the theorem is as simple as 2 +
2 = 4.
2. The following
figure was used by President Garfield in proving the Pythagorean theorem.
His method is based on the fact that the area of the trapezoid ACED is
equal to the sum of the areas of the three right triangles ACB, ABD,
and BED.
Prove the Pythagorean
theorem using President Garfield’s method.
3. A Hindu mathematician
named Bhaskara used the following figure to prove the Pythagorean theorem
by showing the sum of the area of the small square and the area of the
four congruent right triangles equal the area of the large square.
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