Transcript Document 7192338

Pythagoras and the
Pythagorean Theorem
Grade 8-9 Lesson
By Lindsay Kallish
Biography of Pythagoras
•Pythagoras was a Greek mathematician and a
philosopher, but was best known for his
Pythagorean Theorem.
•He was born around 572 B.C. on the island of
Samos.
• For about 22 years, Pythagoras spent time
traveling though Egypt and Babylonia to educate
himself.
•At about 530 B.C., he settled in a Greek town in
southern Italy called Crotona.
•Pythagoras formed a brotherhood that was an
exclusive society devoted to moral, political and
social life. This society was known as Pythagoreans.
Biography of Pythagoras
• The Pythagorean School excelled in many
subjects, such as music, medicine and
mathematics.
• In the society, members were known as
mathematikoi.
• History tells us that this theorem has been
introduced through drawings, texts,
legends, and stories from Babylon, Egypt,
and China, dating back to 1800-1500 B.C.
• Unfortunately, no one is sure who the true
founder of the Pythagorean Theorem is.
But it does seem certain through many
history books that some time in the sixth
century B.C., Pythagoras derives a proof
for the Pythagorean Theorem.
Venn Diagram Homework Assignment
•
•
http://www.arcytech.org/java/pythagoras/history.html
http://www-groups.dcs.stand.ac.uk/~history/Biographies/Pythagoras.html
1st Website
2nd Website
Can Any Three Numbers Make A Triangle????
Side lengths (cm)
Area of square (cm²)
Sum of the
areas of
smaller
squares (cm²)
Sum of the areas
of smaller
squares (cm²)
Angle name of
the triangle
Column 1
Column 2
Column 3
Column 4
Column 5
a
b
c
a²
b²
c²
a² + b²
a² +b² ? c²
(>,<,=)
Acute, right,
obtuse
3
5
7
9
25
49
34
<
obtuse
3
4
5
9
16
25
25
=
right
Write a sentence to describe the relationship between the sum of the areas
of smaller and middle size squares compared to the area of the largest
squares for:
1) An acute triangle
2) A right triangle
3) An obtuse triangle
The Pythagorean Theorem
• The sum of the squares of
each leg of a right angled
triangle equals to the
square of the hypotenuse
a² + b² = c²
Many Proofs of the Pythagorean Theorem
Euclidean Proof
• First of all, ΔABF = ΔAEC by SAS. This is because,
AE = AB, AF = AC, and
BAF = BAC + CAF = CAB + BAE = CAE.
• ΔABF has base AF and the altitude from B equal
to AC. Its area therefore equals half that of
square on the side AC.
• On the other hand, ΔAEC has AE and the
altitude from C equal to AM, where M is the
point of intersection of AB with the line CL
parallel to AE.
• Thus the area of ΔAEC equals half that of the
rectangle AELM. Which says that the area AC²
of the square on side AC equals the area of the
rectangle AELM.
• Similarly, the are BC² of the square on side BC
equals that of rectangle BMLD. Finally, the two
rectangles AELM and BMLD make up the
square on the hypotenuse AB.
• QED
•
http://www.sunsite.ubc.ca/LivingMathematics/V001N0
1/UBCExamples/Pythagoras/pythagoras.html
Many Proofs of the Pythagorean Theorem
Indian Proof
• Area of the original square is A = c²
•
Looking at the first figure, the area of
the large triangles is 4 (1/2)ab
•
The area of the inner square is (b-a) ²
•
Therefore the area of the original
square is A=4(1/2)ab + (b-a) ²
•
This equation can be worked out as
2ab + b² - 2ab + a² = b² + a²
•
Since the square has the same area no
matter how you find it, we conclude
that
A = c² = a² + b²
Many Proofs of the Pythagorean Theorem
• Throughout many texts, there are about
400 possible proofs of the Pythagorean
Theorem known today.
• It is not a wonder that there is an
abundance of proofs due to the fact that
there are numerous claims of different
authors to this significant geometric
formula.
• Specifically looking at the Pythagorean
Theorem, this unique mathematical
discovery proves that there is a limitless
amount of possibilities of algebraic and
geometric associations with the single
theorem.
• http://www.cut-theknot.org/pythagoras/index.shtml
Connection to Technology
• Geometer’s SketchPad
– Students can see the
Pythagorean Theorem work
using special triangles with
45-45-90 degree angles and
30-60-90 degree angles
Pythagoras Board Game
Rules:
• To begin, roll 2 dice. The person with the highest sum
goes first.
•
To move on the board, roll both dice. Substitute the
numbers on the dice into the Pythagorean Theorem
for the lengths of the legs to find the value of the
length of the hypotenuse.
•
Using the Pythagorean Theorem a²+b²=c², a player
moves around the board a distance that is the integral
part of c.
•
For example, if a 1 and a 2 were rolled, 1²+2²=c²;
1+4=c²; 5=c²; Since c = √5 or approximately 2.236, the
play moves two spaces. Always round the value down.
•
When the player lands on a ‘?’ space, a question card
is drawn. If the player answers the question correctly,
he or she can roll one die and advance the resulting
number of places.
•
Each player must go around the board twice to
complete the game. A play must answer a ‘?’ card
correctly to complete the game and become a
Pythagorean
Pythagoras Board Game
What are the
lengths of the legs
of a 30-60-90
degree triangle
with a hypotenuse
of length 10?
Answer: 5 and 5√3
If you hiked 3
km west and the
4 km north, how
far are you from
your starting
point?
Answer: 5 km
The square of the
______ of a right
triangle equals
the sum of the
squared of the
lengths of the two
legs.
Answer:
hypotenuse
Find the missing
member of the
Pythagorean triple
(7, __,, 25).
Answer: 24
What is the length
of the legs in a 4545-90 degree right
triangle with
hypotenuse of
length √2?
Answer: 1
Using a²+b²=c,
find b if c = 10
and a = 6
Answer: b=8²
True or false?
Pythagoras lives
circa A.D. 500
Answer: false (500
B.C.)
Have the person to
your left pick two
numbers for the legs
of a right triangle.
Compute the
hypotenuse
Can an isosceles
triangle be a right
triangle?
Answer: yes
Pythagoras was
of what
nationality?
Answer: Greek
Is (7, 8, 11) a
Pythagorean
triple?
Answer: no
How do you spell
Pythagoras?
The Pythagorean
Theorem is
applicable for what
type of triangle?
Answer: a right
triangle
What is the
name of the
school that
Pythagoras
founded?
Answer: The
Pythagorean
School
True or false?
Pythagoras
considered
number to be the
basis of creation?
Answer: true
True or false?
Pythagoras
formulated the only
proof of the
Pythagorean
Theorem?
Answer: false (there
are about 400
possible proofs)
References:
DeLacy, E. A. (1963). Euclid and geometry
(2nd ed.). USA: Franklin Watts, Inc.
Ericksen, D., Stasiuk, J., & Frank, M. (1995).
Bringing pythagoras to life. The
Mathematics Teacher, 88(9), 744.
Gow, J. (1968). A short story of greek
mathematics. New York: Chelsea
Publishing Company.
Katz, V. (1993). A history of mathematics (2nd
ed.). USA: Addison Wesley Longman, Inc.
Swetz, F. J., & Kao, T. I. (1977). Was
pythagoras chinese? an examination of
right triangle theory in ancient china. USA:
The Pennsylvania State University.
Veljan, D. (2000). The 2500-year-old
pythagorean theorem. Mathematics
Magazine, 73(4), 259.