Using Technology and MediaRich Platforms to help teach the Pythagorean Theorem
Using Technolog y and MediaRich Platforms to help teach the Pythagorean Theorem
Hui Fang Huang Su 0
0 Nova Southeastern University
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Using Technolog y and MediaRich Platforms to help teach the
Pythagorean Theorem
Authors
Hui Fang Huang Su, Denise Gates, Janice Haramis, Farrah Bell, Claude Manigat, Kristin Hierpe, and
Lourivaldo Da Silva
This article is available in Transformations: http://nsuworks.nova.edu/transformations/vol2/iss2/4
Title: Using technology and mediaRich platforms to help teach the Pythagorean theorem
The history of mathematics is quite extensive, dating back to ancient times and
elapsing several centuries. Many great mathematicians, philosophers, scientists, and scholars
have contributed to mathematics on the subject we see today. For the purpose of this research
project, we have chosen to highlight the great mathematician Pythagoras and his famous
Pythagorean Theorem. We investigate the history of the Pythagorean Theorem, relationships
among Pythagorean numbers, and unusual proofs of the Theorem. This paper provides a
suggestive lesson plan, activity, and presentation for classroom use.
History of the Pythagorean Theorem
One of the most wellknown theorems in our culture that many may remember
from their earlier high school days is the Pythagorean theorem (a2 + b2 = c2). Over the centuries,
there have been many names for the Pythagorean theorem, including the Theorem of Pythagoras,
the Hypotenuse theorem, and Euclid I 47 (since it is listed as Proposition 47 in Book I of
Euclid’s Elements). The Pythagorean theorem has had quite an impact, and there are several
stories on its advancement
(Maor, 2007)
.
In The Ascent of Men, Jacob Bronowski stated: “To this day, the theorem of
Pythagoras remains the most important single theorem in the world of mathematics” (p.60). The
Pythagorean theorem has influenced mathematics by virtually permeating every branch of
science and playing a central role in various mathematical applications, such as geometry.
Moreover, the classical theorem has over four hundred existence proofs, and it continues to
grow. In fact, on the original list of proofs, there is one from Albert Einstein from age twelve
(Maor, 2007)
.
Upon further research on the history of the Pythagorean theorem, once can
conclude that the Pythagorean formula was discovered independently, in some sense, by almost
every ancient culture (Mackenzie, 2012). The Pythagorean theorem evolution goes back to over
2,500 years, to when Pythagoras theoretically first proved it. Notably, Pythagoras was not the
first to discover the theorem, as recognized as the Babylonians, and probably the Egyptians and
Chinese, at least, a thousand years ahead of him (Moar, 2007).
In 2000 BCE, Ancient clay tablets denote that the Babylonians created rules for the
Pythagorean triplets. In ~1700 BCE, a famous Babylonian tablet called the Plimpton 322
includes a list of Pythagorean triplets (ex. 345). Discovering the tablet around 1945, which is
currently on display at Yale University
(Allen, 2003)
. Research proves that Pythagoras studies
in Babylonia and might have learned the theorem there (Mackenzie, 2012). In 1700 BCE, Pierre
de Fermat studied the Pythagorean theorem and conjectured that there were no solutions when n
was greater than two. Wittingly, he left no proof and only wrote in the hand left margin that he
knew this was not possible, but did not have enough room to write it. This conjecture was
named the “Fermat’s Last Theorem” and was not proved until 1993 by Andrew Wiles, from
Princeton University
(Allen, 2003)
.
Modern algebra was on the rise in 1600 BCE, and the Pythagorean theorem
assumed it’s now wellknown algebraic form
(Maor, 2007)
. In 1500 BCE, the Pythagorean
theorem states: “The square of the lengths of the hypotenuse of the right triangle is equal to the
sum of the squares of the legs” (Historical Timeline). Between 800 BCE 600 BCE, an ancient
Indian Mathematician discussed the Pythagorean theorem in Sulbasutras. In 569 BCE,
Pythagoras was born on the island of Samos in Greece. The Pythagorean theorem derived from
the ancient Greek mathematician, Pythagoras. Many believe Pythagoras presented the first proof
of the theorem, although it was discovered long before him
(Allen 2003)
.
Furthermore, theories state that Pythagoras may have even traveled to India around
500 BCE, visiting eastern regions such as Egypt and Phoenicia. Supposedly, Pythagoras learned
from the Phoenicians and Egyptians and their priests and acquired some early notions about
mathematics (Aczel). “The Egyptians must have use this formula [a2 + b2 = c2] or they couldn’t
have built their pyramids, but they have never expressed it as… as useful theory,” from Joy
Hakim, The Story of Science (page 78). Unfortunately, our knowledge of the Egyptians is not
extensive due to the fragile mediums of evidence. The Babylonians wrote on clay tablets, which
are virtually indestructible, whereas Egyptians wrote on delicate papyrus. The Egyptian
monument, the “Great Pyramid of Cheops” is 756 feet on each side with a height of 481 feet.
This example will require scientific information, which could include the Pythagorean theorem.
However, did it really? The Rhind Papyrus consists of eightyfour problems and also contains
Pyramid problems, but fails to reference them to the Pythagorean theorem (nor directly or
indirectly). Another hypothesis states that Egyptians used a rope tied at equal intervals to
measure distances, which could have led them to discover the 345 triangle. There is no
evidence to support this theory
(Maor, 2007)
.
Around 500 BCE 200 BCE, the Chinese treatis, Chou Pei Suan Ching, offers a
statement and demonstration of the Pythagorean theorem. Hippocrates of Chios (470410 BCE)
lived well before Euclid’s period, and researchers say he generally knew of the Pythagorean
theorem, as he stated in Proposition 31 of Book VI: “In rightangled triangles, the figure on the
side opposite the right angle equals the sum of the similar and similarly described figures on the
sides containing the right angle”
(Alsina, 2011)
. In 300 BCE, Euclid’s Elements is published and
includes a proof of the Pythagorean theorem (Mackenzie, 2012).
The Pythagorean theorem is one of the most wellknown theorems in geometry, the
most popular property of right angles, and contains more proofs than any other in mathematics.
In 1876, a Pythagorean theorem proof was published in the New England Journal of Education
(Volume 3, page 161), written by James Abram Garfield (18311881). During the time, Garfield
was a member of the Ohio United States House of Representatives, until 1880, when he was
elected the twentieth President of the United States. Sadly, he was assassinated only four months
after being elected
(Alsina, 2011)
.
In 1895, Lewis Carrol commented on the beauty of the Pythagorean theorem: “It is
as dazzlingly beautiful now as it was in the day when Pythagoras first discovered it.” In 1927,
Elisha S. Loomis (18521940), a mathematics teacher from Ohio, collected and wrote up three
hundred seventyone proofs in The Pythagorean Preposition. In 2004, the journal Physics
World, awarded the Pythagorean theorem 4th place (out of 20), as voted by readers, for the
twenty most beautiful equations in science
(Maor, 2007)
.
The Pythagorean theorem has evolved and impacted numerous fields of math and
science over the past decades. A wide variety of diverse cultures has contributed to its history.
The Pythagorean theorem has more proofs than any other in mathematics, and each one offers
something different. Today, we can only assume, Pythagoras, in all probability, supplied the
first proof, and gets the name of this universal theorem.
Unusual relationships that exist among Pythagorean numbers
Did you know there is more to Pythagoras than the Pythagorean Theorem?
Pythagoras and his followers believed that all relationships to numbers could relate to and
account for geometrical properties. Pythagoras and his supporters studied the properties of
numbers that are familiar to us today. They studied even and odd numbers, prime numbers, and
square numbers. Also, Pythagoras and his supporters believed that numerical properties were
masculine or feminine, perfect or incomplete, and beautiful or ugly. Odd numbers were known to
be masculine while even numbers were reflected feminine because they were weaker than the
odd numbers. Furthermore, the odd numbers were considered the master numbers, because an
odd plus and even always produces an odd. Also, two evens can never produce and odd number,
while two odds always produce an even number.
Initially, Pythagoreans believed in the existence of whole numbers only; however,
as time passed, they proved the existences of incommensurables. Incommensurables are what we
know today as irrational numbers. Pythagoras and his followers also used patterns of dots to
represent numbers. The resulting figures, created by the patterns of dots, are known as figure
numbers. For example, nine dots can be arranged in three rows with three dots per row, creating
the picture of a square. Similarly, ten dots can be organized into four rows, containing one, two,
three, and four dots per row so as to form a triangle. Develop from the various figure numbers,
Pythagoras, and his followers further derived relationships between numbers. They discovered
that a square number could be subdivided by a diagonal line, into two triangular numbers; this is
why we can say that square numbers are always the sum of two triangular numbers. For
example, the square number of twentyfive is the sum of the triangular number ten and the
triangular number fifteen.
Unusual Proofs of the Pythagorean Theorem
According to the writer A. Bogomolny, there are so many unusual ways to proof
the Pythagorean Theorem. By incorporating those unique strategies to teach it in high school,
teachers will be able to engage and lead students to learn the Pythagorean Theorem in a
remarkable way. For instance, one of the cited proofs makes use of the area of the big square
KLMN is b². The square splits into four triangles and one quadrilateral:
The area of the big square KLMN is b². The square divisions into one quadrilateral and four
triangles:
b² = Area(KLMN)
= Area(AKF) + Area(FLC) + Area(CMD) + Area(DNA) + Area(AFCD)
= y(a+x)/2 + (bax)(a+y)/2 + (bay)(bx)/2 + x(by)/2 + c²/2
= [y(a+x) + b(a+y)  y(a+x)  x(by)  a·a + (bay)b + x(by) + c²]/2
= [b(a+y)  a·a + b·b  (a+y)b + c²]/2
= b²/2  a²/2 + c²/2. ((Bogomolny, A., 2012)
Visual model of Pythagorean Theorem
The following model is a visualization of the Pythagorean Theorem: 3^2 + 4^2 = 5^2 9+16=25 25=25
If you put three and four together, it will not equal five, so you can’t put the original two squares
together, to get the big square. The best way to get the two smaller squares to fit into the big
square when they are combined is to square the number of tiles in each. So, three squared equals
nine, and four squared equals sixteen, and nine plus sixteen equals twentyfive which will fit on
the large square when combined.
Or this one:
5^2+12^2=13^2
25+144=169
169=169
If you put five and twelve together, it will not equal thirteen, so you can’t put the original two
squares together, to get the big square. The best way to get the two smaller squares to fit into the
big square when they are combined is to square the number of tiles in each. So, five squared
equals twentyfive and twelve squared equals one hundred and fortyfour, and twentyfive plus
one hundred and fortyfour equals one hundred and sixtynine, which will fit on the large square
when combined. You can use any four sided figure you want to put together and form a right
triangle in the middle.
Concluding Remarks
This record of the Pythagorean Theorem is far from a full documentation of the
brilliant work of Pythagoras. It does, however, highlight some of the most important discoveries
regarding this Theorem. It is imperative to note that very few mathematicians have come close
to, or would be able to discover the ideas they have, without it. The attached lesson plan, activity,
and presentation aim to deliver this information in a studentfriendly format. Through this lesson
plan, it is our goal that students develop an understanding not only of the Theorem itself but of
its place in history and importance in the development of modern day mathematics.
Lesson Plan
Grade Level: 6th  9th Grade
Lesson Duration: 90 Minutes
Overview:
Lesson Plan: Applying the Pythagorean theorem
This lesson allows students to experience the Pythagorean theorem through the use of a
mediarich interactive learning platform. Through an handson approach, students will deduce that the
angle classification of a triangle is determined by its side measurements. Students will find the
perimeter of polygons on the coordinate plane by triangulation. They will also integrate their
reading, writing, and mathematical skills during this learning experience.
Objectives: Student will be able to:
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Materials:
Technology Component:
Find the square of a number using the area of a square.
Find a square root of a number by using the length of the side of a square of the same
area.
Discover the Pythagorean theorem using the handson approach.
Find the sides of a right triangle using the Pythagorean theorem.
Conclude that the Pythagorean theorem is a relationship between areas.
Use the Pythagorean theorem to find the right triangle and the length of an unknown side.
PowerPoint
Copies of a Pythagorean assignment sheet (one per student)
Set of calculators
Graphic paper or dot paper
Set of rulers
Construction Paper Triangle
• IPAD Cart
1. Introduction to the lesson starts with a brief lecture on the history of Pythagorean
Theorem and righttriangle. The concepts of area and length are discussed, stressing the
fact that the Pythagorean Theorem is fundamentally a relationship between areas.
2. Students will watch a two minutes demo video on the Pythagorean Theorem, and students
will then share with their partners, two facts about Pythagorean Theorem and two facts
about two mathematicians’ proofs of the Pythagorean Theorem. They also had to discuss
and ask each other questions about their evidence by the thinkpairshare method.
3. Write Pythagorean Theorem equation down on the Promethean board and have a
meaningful discussion about the different parts of the theorem. Comprehensively explain
what a, b and c means in the equation. Explain that a and b and the legs of the triangle
while c is the hypotenuse of the triangle.
4. The classroom formation is arranged into groups. Each group is given four identical right
triangles labeled with hypotenuse and legs. The students are asked to organize their
triangles into a square with the four hypotenuses forming the perimeter of the square.
5. Next, students are guided through the proof by being asked a series of questions about the
arrangement of triangles.
6. Allow 2025 minutes for engaging discussion. Regroup the class together and call pairs
of students to clarify how they will solve the problems. Feel free to ask openended
questions (make sure you are not judging or communicating to students they are wrong or
right.) If time is not permitted for all the problems discussion, the lesson will be extended
to the next day for three minutes opening activity. The lesson could divide over two days.
There are many ideas to discuss.
Accommodation:
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Student with specific learning disability in reading, assist the student by reading the
directions and make sure that he or she understands.
For a gifted and talented student in math and reading, challenge the student with story
problems involving right triangles and using the Pythagorean Theorem.
Student with attention deficit hyperactivity disorder (ADHD) and emotional disabilities,
allow the student to work in a group setting so that the student can have a positive role
model; in fact, teacher believes that students will stay on task better if his/her peers are on
task. The teacher would allow the students to walk around during homework time for a
short break.
At the beginning of the intermediate levels of proficiency, English Language Learners
should be exposed to samples of completed working assignments to model the correct
format.
Assessments:
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During the activity, the teacher will implement an informal evaluation by circulating
through the classroom, asking questions, and providing feedback.
Exit Ticket: Students will take Quiz over Pythagorean Theorem. Only If the time permits.
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CrossCurricular Connection: In eighthgrade world history, students study ancient Greek
philosophers. Students will be able to use the succeeding websites to research Pythagoras.
They will create a slideshow presentation to present five interesting facts about the
philosopher.
Student Sample Work:
Construction Paper Triangle or Triboards
This project used a 9th grade Math class of twentysix students, two of which are English
language learners. It is a high achiever/general class setting, which is a doublecoded class with
varying ability levels. The lesson used for this project is entitled “Applying the Pythagorean
Theorem.” In this lesson, the students examined the Pythagorean theorem to realize that in order
to solve problems, the side measurement is critical in determining the angle measure
classification of a triangle. When squaring sides, they can predict whether triangles will be right,
obtuse, or acute. Using the Internet, the students will prove the Pythagorean Theorem, and that
will enable them to use it to solve problems. Lastly, the students will demonstrate their
knowledge of the Pythagorean Theorem by determining the perimeter of polygons on the
coordinate plane by triangulation. This activity utilizes reading, writing, and mathematical skills
in an interdisciplinary format.
At first, the activity begins with a brief lecture introducing righttriangles and the history
of Pythagorean Theorem. The concepts of length and area are examined, stressing the fact that
the Pythagorean Theorem is fundamentally a relationship between areas. Then, the students
watched a demo instruction video on the Pythagorean theorem. Students then shared with their
partners, two facts about Pythagorean Theorem and two facts about two mathematicians’ proofs
of the Pythagorean Theorem. They also had to discuss and ask each other questions about their
evidence using the thinkpairshare method. The teacher walked around and listened to various
responses and at the same time poses questions to elicit conversation.
The teacher changed questions around to make them openended questions. For example,
“How do you think we are going to solve for the short side of the triangle?” One student
responded, “the Pythagorean theorem is going to be our life jacket to save us.” The teacher then
asked, “What makes it a life jacket to saving us?” The student’s responses made the teacher
cleared up any misconceptions. Closed questions deny students the opportunity of meaningful
conversation. In looking back at the video, the teacher realized that her students will also benefit
from a lesson on open and closed responses. If they are made more aware of this, they would be
able to stimulate discussions that would provide opportunities for meaningful practice of the
language of math. Some of the questions asked by students were closed questions. They were
asking questions that elicited yes or no responses. The teacher modeled openended questions to
the groups she visited; however, they need more practice as students were not able to grasp the
concept fully.
The teacher made sure to visit her English language learner’s (ELL) student group. She
placed them in the same small group for easy access. During one of her visits to their group, she
noticed they were struggling with some words. So, she read the directions and the problem to
assist with vocabulary comprehension. Also, she posed simple questions about the arrangement
of triangles to guide them to answering the question. It is imperative that teacher’s repeat phrases
and sentences so that ELLs can fully comprehend what is being said. Looking back at the video,
she could see the “light bulb” come on for them as she repeated the sentences.
Discourse analysis has helped the teacher to look for recurring patterns in her questioning
style. By analyzing how particular questions impacted her students’ responses, she made
necessary changes, and she is now asking more challenging and thoughtprovoking questions to
promote active and meaningful dialogue. For this reason, the final activity was designed to be
independent group practice that will enable her to observe how well the students understood the
main ideas covered in the lesson. It had them complete some procedural problems using the
Pythagorean Theorem. Students will be giving problems that they need to apply relevant
properties that they learned in the lesson. There was a representation of practice problem that
appropriately dealt with everything that was covered, which give the teacher a reasonably
accurate assessment of their knowledge. Along with the discussions during class allowed the
teacher to assess the levels of understanding for each student.
Aczel, A. D. (n.d.). Finding zero: A mathematician's odyssey to uncover the origins of numbers.
Mackenzie, D. (2012). Universe in Zero Words: The Story of Mathematics As Told Through
Equations. Princeton, NJ, USA: Princeton University Press. Retrieved from
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Allen , G. D. ( 2003 ). Babylonian Mathematics. Retrieved February 10 , 2016 , from http://www.math.tamu.edu/~dallen/masters/egypt_babylon/babylon.pdf
Alsina , C. , & Nelsen , R. B. ( 2011 ). Dolciani Mathematical Expositions , Volume 45 : Icons of Mathematics: An Exploration of Twenty Key Images . Washington, DC, USA: Mathematical Association of America. Retrieved from http://www.ebrary.com
Atkins , William Arthur; Koth, Philip Edward . "Pythagoras." Mathematics . 2002 . Retrieved February 10, 2016 from Encyclopedia .com: http://www.encyclopedia.com/doc/1G2 3407500247 .htm
A Brief History of the Pythagorean Theorem . (n.d.). Retrieved February 10 , 2016 , from http://www.geom.uiuc.edu/~demo5337/Group3/hist.html
Bogomolny , A. ( 2012 ). Pythagorean theorem and its many proofs from interactive mathematics miscellany and puzzles . Retrieved from http://www.cuttheknot.org/pythagoras/
Maor , E. ( 2007 ). The Pythagorean theorem: A 4 , 000  year history. Princeton, NJ: Princeton University Press.
Su , H. F. , Ricci , F. , Mnatsakanian , M. ( 2016 ). Critical Thinking as a Mathematical ProblemSolving Strategy: Making the Case: Classroom Scenario. Transformations: A Publication of the Florida Association of Mathematics Teacher Educators (FAMTE) . Reprinted from Dimensions in Mathematics Journal. Pages 46  53 . Volume 1 , Issue 1. Winter 2016 .
Su , H. F. , Ricci , F. , Mnatsakanian , M. ( 2016 ). Mathematical Teaching Strategies: Pathways to Critical Thinking and Metacognition . International Journal of Research in Education and Science (IJRES) . Volume 2 , Issue 1 , Winter 2016 , pp. 190  200 . ISSN: 2148  9955 .
Su , H.F. , Sico , J. , Su , T.C. ( 2016 ). A Simple approach to Algebra and Higher Order Thinking Skills (Second Edition) . An Amazon.com publication. January 28 , 2016 . ISBN: 9781523758364 .