Real+Algebraic+Irrational+Numbers

=Real Algebraic Irrational Numbers=

==**Depict Irrational Lengths.** == with a side of rational length (example: square of side = 3/2). (a) Drawing a right triangle whose three sides are all rational numbers (recall special triangles from Geometry). (b) Drawing a right triangle whose hypotenuse is not a rational number (example, square root of 2).
 * 1. Illustrate at what it means to say that the rational numbers are //closed// under exponentiation by depicting a perfect square
 * You may draw your figure by hand, scan your drawing, and upload the file into the cell below.
 * Include the names of group members participating in this section of the lesson. ||
 * 2. Illustrate at what it means to say that the rational numbers are //not closed// under square roots by:
 * 2. Illustrate at what it means to say that the rational numbers are //not closed// under square roots by:
 * You may draw your triangles by hand, scan your drawing, and upload the file into the cell below.
 * Include the names of group members participating in this section of the lesson. ||

**Rational Powers**
Recall that (b^3)^2 = b^(3*2)=b^6. Since multiplication of powers is related to exponentiation, "canceling" a power is related to the multiplication's inverse operation. For example, the square root of 9 can be written (3^2)^(1/2).
 * 1) Write the cube root of //x// using a fractional exponent.
 * 2) Simplify 9^(3/2) without using a calculator [recall that 3/2 = (1/2)*3].
 * 3) Explain why [9^(1/2)]^3 = [9^3]^(1/2)

Identify group members working on this section of the lesson.

**Write a proof:**
Upload your proof in the cell below. ||
 * Write an indirect proof that the square root of 2 is irrational. You should be able to find this information online.

==**The Golden Ratio, Phi ** ==

The rectangle depicted above is constructed from a square of side = 1. Bisect a side of the square to create two rectangles 1/2 x 1 in size. Answer the following questions below. Identify group members participating in this section of the lesson.


 * 1) Use the Pythagorean Theorem to find the length of the diagonal of the rectangles. (Show work.)
 * 2) The diagonal found above is used as the radius of a circle whose center is at the bisection point, as depicted above. What is the length of the longer side of this golden rectangle?
 * 3) Research evidence of the Golden Rectangle and the facade of the Parthenon. Consider using the following site: []
 * 4) When and where was the Parthenon built?

Create an attractive illustration (8.5"x11") depicting this Pythagorean construction for the Golden Ratio, Phi. You might want to include a picture of the Parthenon. This illustration will be added to the class's timeline/map in the back of the classroom. Cite any sources used and identify group members participating in this section of the lesson. ==**The Greek Sculptor and Architect Phidias** == Research Phidias. Consider using the following site: [] At a minimum, answer the following questions in the space below:
 * 1) When and where did he live?
 * 2) What is he known for?
 * 3) What is the connection between his name and the Golden Ratio, Phi?

Use the guidelines provided in class to collaborate using Google Docs to create an attractive trading card (half of an 8.5"x11" sheet of paper) including a picture and a brief synopsis of Phidias' life, including a significant accomplishment. Cite any sources used and identify group members participating in this section of the lesson.