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Big Ideas:
Behind every measurement formula lies a geometric result.
This lesson introduces the Pythagorean Theorem using a geometric model that students rearrange to visually 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 and rearranging the geometric shapes helps students connect the visual representation of the geometric shapes with the numeric and symbolic representation of the algebraic formula.
In this lesson, students will dissect a square with sides of length a + b by cutting it according to Bhaskara II's model. Next, they will rearrange the pieces, which include four congruent triangles with bases of length a and heights of length b and one square with side lengths of c. This provides the visual proof of the Pythagorean Theorem. Then, using the formulas for the area of triangles and squares, students will compare the area of the original square with the new square. After some algebraic manipulation, students will have developed the Pythagorean Theorem.
You may wish to share the following historical information with your class when teaching this lesson. Bhaskara II was a prominent twelfth century astronomer and mathematician who enjoyed playing with mathematical ideas and numbers. He wrote about arithmetic, algebra, mathematics of the planets, and spheres. In these works, he explained the decimal number system and principles of differential equations long before Newton and Liebniz discovered them. He even wrote mathematical problems in the form of poems. Bhaskara II knew about the relationship between the squares of the sides of a right triangle and wrote two geometric proofs that illustrate the relationship. This lesson uses one of his proofs for the task and the other geometric proof as an extension task.
Special Materials:
Paper models of Bhaskara II's square
Graph paper or isometric dot paper
Scissors
Colored pencils