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Big Idea:
Relationships can be described and patterns predicted for mathematical shapes and numbers that repeat.
This lesson builds on students' work using the Pythagorean Theorem to find the unknown side lengths of right triangles. The task focuses on finding patterns in contiguous right triangles and on the value of irrational numbers that are often the result of applying the Pythagorean Theorem.
Students start with an isosceles right triangle with legs of 1 unit and find the hypotenuse (an irrational number). This hypotenuse becomes a leg of an adjoining right triangle whose other leg is still 1 unit. They find the hypotenuse of this triangle and continue to build right triangles using the previous hypotenuse and the Pythagorean Theorem.
The mathematical concept in this lesson is that relationships can be described and patterns predicted for mathematical shapes and numbers that repeat. As more elements of the pattern are known, they can be used to predict other elements. This lesson also gives students a visual model for understanding and relating irrational numbers to the perfect squares they lie between. It gives students a way to compare the size of irrational numbers without using a number line.
You may wish to share the following historical information with your class when teaching this lesson. Theodorus was a 5th century B.C., Greek mathematician, who was a member of the society of Pythagorus and a tutor of Plato. He lived in Cyrene, which is now Shahhat, Libya. What we know about Theodorus and his mathematical contributions comes from what Plato wrote about him in Plato's dialogues.
Vocabulary: rational number, irrational number, Pythagorean Theorem, right triangle, isosceles triangle, leg, hypotenuse, spiral, radical, simplest radical form, square root, radicand
Special Materials:
ruler
protractor
plain paper (8 1/2 x 11 or larger)
isometric dot paper (optional)