Transcript Reasoning with Area Principles Math Alliance December 7, 2010 Session Goals • To explore the concept of area • To see the relation between the.

Reasoning with Area
Principles
Math Alliance
December 7, 2010
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
Area of a Parallelogram
Area of a Triangle
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?
Where Do Area Formulas Come
From?
• What is the area of a triangle?
• A parallelogram?
• How do you know?
More Area Practice
Area of a Square
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.
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.
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-theknot.org/pythagoras/index.shtml
Proofs of the Pythagorean Theorem
• Here is proof #9 in its entirety:
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?