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Big Ideas:
Geometric shapes and diagrams help to visualize any measurement formula. When we decompose and rearrange geometric shapes, we can connect the visual geometric shapes with abstract and symbolic representations of the algebraic formula.
This lesson develops the geometric proof of the Pythagorean Theorem developed by James Garfield, the 20th President of the Unites States. Students use a trapezoid made from two congruent right triangles and an isosceles right triangle to represent the concept that there is an equivalent relationship between the sum of the squares of the legs of a right triangle and the square of the hypotenuse. Decomposing the trapezoid into the three right triangles helps students connect the visual representation of the geometric shapes with the algebraic formula. When students compare the area of a trapezoid to the area of its composite figures, they get the Pythagorean Theorem in a slightly different form. The final result is that half of the sum of the squares of the legs of a right triangle is equal to half of the square of the hypotenuse. Garfield's geometric model is basically half of Bhaskara II's model.
You may wish to share the following historical information with your class when teaching this lesson. Before James Garfield became President in 1881, he was a member of Congress for 20 years. During that time, he discussed his trapezoid model of the Pythagorean Theorem with other members of Congress and explained how it would prove the Pythagorean Theorem. He even published his proof in the New England Journal of Education in 1876. There is no evidence that Garfield's proof added votes to his election or that it had any connection to his assassination in 1881, six months after his election.
Vocabulary: hypotenuse, legs, right angle, right triangle, trapezoid, congruent, contiguous, isosceles
Special Materials:
Diagram of Garfield's model.
graph paper (optional)
isometric dot paper (optional)