1. Reasoning with Area Principles Math Alliance December 7, 2010

2. Session Goals To explore the concept of area To see the relation between the concept of area and formulas for areas of basic shapes To discover an important theorem about area

3. Area of a Parallelogram

4. Area of a Triangle

5. Properties of Area What properties of area did you use in finding the areas of the parallelogram and the triangle? – The “moving property”: the area of a shape is not changed if the shape undergoes a rigid motion – The “combining property”: the total area of two (or more) non-overlapping shapes is the sum of their individual areas How do these properties relate to the idea of area as the number of square units required to cover a shape?

6. Where Do Area Formulas Come From? What is the area of a triangle? A parallelogram? How do you know?

7. More Area Practice

8. Area of a Square

9. Developing a Conjecture About Areas Draw a right-angled triangle near the center of a sheet of grid paper. – You should draw the triangle with two of its sides parallel to grid lines. Draw a square on each side of the triangle.

10. Developing a Conjecture About Areas Use the area properties we discussed earlier— especially the moving and combining principles—to find the areas of your 3 squares. Compare your results with those of people sitting near you.

11. Proofs of the Pythagorean Theorem You can find over 80 proofs of the Pythagorean Theorem at the Cut the Knot website: – http://www.cut-the-knot.org/ http://www.cut-the-knot.org/ – http://www.cut-the- knot.org/pythagoras/index.shtml http://www.cut-the- knot.org/pythagoras/index.shtml

12. Proofs of the Pythagorean Theorem Here is proof #9 in its entirety:

13. The Converse to the Pythagorean Theorem What is the converse statement to the Pythagorean theorem? Is the converse true? In other words, is the converse to the Pythagorean theorem also a theorem? How do you know?