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?
