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Big Ideas:
Some sequences of geometric objects change in predictable ways.
The names of geometric shapes draw attention to their properties which can be used to discover geometric and algebraic patterns related to the shapes.
This lesson uses the Scarecrow from the Wizard of Oz and his mistake about the Pythagorean Theorem to help students discover the special relationships between the lengths of the sides of an isosceles right triangle and the length of the sides of a triangle with angle measurements of 30-60-90 degrees. Students will compare the length of the hypotenuses to the length of the legs of various isosceles right triangles to find a pattern that can be used to determine the hypotenuse or leg of any isosceles right triangle. They will then look for a pattern in the length of the sides of 30-60-90 degree triangles by using the altitude of an equilateral triangle to make two 30-60-90 degree triangles and the Pythagorean Theorem to find the missing leg. Students will simplify radicals and decompose numbers into factors that can be irrational numbers to facilitate recognizing and using the pattern. This task emphasizes that naming an object draws attention to the properties of the object and those properties can be used to discover geometric and algebraic patterns related to the object.
Vocabulary: isosceles, hypotenuse, irrational numbers, radical, radicand, right triangle
Special Materials:
dot paper or graph paper
protractor
ruler