1. The proofs of the Early Greeks 2800 B.C. – 450 B.C.

2. Pythagorean Theorem  Write an equation illustrating the relationship between the area of the rectangles, the area of the white smaller square, and the area of the large square  Write an equation illustrating the relationship between the smaller squares, the rectangles and the larger square.  Use transitivity and the subtraction property of equality.

3. Pythagorean formula for integer solutions Exercise 1: Find a right triangle with side length equal to 9. Exercise 2: Find a right triangle with side length 14.

4. Irrational Numbers! Relatively prime = no common factors

5. Irrational Numbers!

7. Greek’s Idea of Algebra Euclid’s Proposition 4 of Book II: If a straight line divided into any two parts, the square on the whole line is equal to the sum of the squares on the two parts together with twice the rectangle contained by the two parts. How does this mean that? 

8. Translating Euclid  Square on the line: A square whose side length is equal to the length of the given segment Square ABDE  Rectangle contained by the two parts: The rectangle whose side lengths are given by the two parts Rectangle ACDF  A straight line is divided: Plot a point somewhere on the line segment. Point C

9. Proposition 5 of Book II If a straight line is divided equally and also unequally, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. Let us translate

10. Proposition 5 of Book II If a straight line is divided equally and also unequally, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line. Create a straight line segment AB Divide it equally : Plot the midpoint. Call this M. And also unequally: Plot a non- midpoint point, Q, on AB. Construct the rectangle contained by the unequal parts. Construct the square on the points M and Q. Add these areas up. The sum should be equal to the square on AM.

11. Does your shape look like this? Verify the postulate!

12. Method of Proportions Find the relationship between x, a, b, and c

13. Method of Proportions