Saturday, October 26, 2013


The Pythagorean Theorem
By Curtis Bustos



The Pythagorean Theorem was named after a Greek mathematician and philosopher named Pythagoras. While Pythagoras has been credited with many mathematical contributions, the Pythagorean Theorem a2 + b2 = c2 is one of his most notable and has made enormous contributions toward the study of trigonometry.

The Pythagorean Theorem states that if you have a right triangle, the square of the hypotenuse is equal to the square of the other two sides of that triangle also known as a2 + b2 = c2. The hypotenuse is the long side of a right-angled triangle and is represented by the letter “c” while the other two sides or legs  “a & b” are each shorter relative to the hypotenuse. The hypotenuse can also be defined as the side opposite to the right angle (90˚ ) of a triangle.





Learning Goal – how the Pythagorean Theorem can be applied to right-angles and their corresponding triangles.
Learning Activity – 5E Instructional model using think-pair-share.

5E Instructional model:

1.     Engagement: draw the students in with a reading.
2.     Exploration: focus on concepts and skills. Mathematical examples are provided.

i.       Introduce a right triangle.  The Pythagorean Theorem is only valid with a right triangle (90˚).  Explain obtuse and acute triangles for comparability.
ii.     Explain terms in relation to a, b, c
A = (legs or right-angle sides)
B = (legs or right-angle sides)
C = Hypotenuse (longest side and opposite to angle). 

a2 +b2 = c2. This theorem states that the sum of the squares of the two right angle sides will always be the same as the hypotenuse.





a2 + b2 = c2 (right angle)
a2 + b2 < c2 (obtuse angle)
a2 + b2 > c2 (acute angle)

iii.    Figure out a2 + b2 = c2 where


                           





and
 


3.     Explanation: Builds on the first two phases by allowing students to determine if they are working with a right triangle?
Use a triangle with 3’, 4’, 5’.







4.     Elaboration: real world application.  Pythagorean theorem is important to drafters, engineers, carpenters, astronomers, x-ray crystallography and many additional important trades. Without x-ray crystallography, our understanding of biological processes would be reduced to a much lower level than we consider standard today. When scientists measure the directions and intensities of x-ray beams, they can deduce chemical structures and biological processes. This would not have been attainable without the famous Pythagorean Theorem.



5.     Evaluation (student understanding):
 What is the diagonal distance across a square?

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