Alison L. Keller
MTHT 202 Project
March 15, 2007
Tracking the Pythagorean Theorem K-8 and Beyond
G1.2.3 – Know a proof of the Pythagorean Theorem and use the
Pythagorean Theorem and its converse to solve multistep problems
The Pythagorean theorem states:
“In a right triangle, the square of the length of the hypotenuse is equal to the sum
of the squares of the lengths of the other two sidesi”
Or
For a right triangle with legs a and b and hypotenuse c,
a +b =c
This is an important concept dealt with in high school geometry, but to prepare students
to be comfortable using the theorem in higher grades, there are building blocks to
learning, and steps to take along the way. There are many implications of the theorem
that lead to complex problems and solutions. You can discover the length of the third side
of a right triangle given two other sides. You can conversely, tell if a triangle is right (or
obtuse or acute) given the lengths of its sides. Further, it can be used to solve multistep
problems and applied to other areas of mathematics and geometry. The following will
give a brief history of its development and use, discuss k-8 standards, which build up to
its use in high school, provide activities that relate to the theorem, and give an example of
its modern application.
Historical Account
Pythagoras of Samos first proved the Pythagorean Theorem around 500 BC. There were,
however, other civilizations aware of and utilizing parts of the theorem well before
Pythagoras was around. Two such cultures were the Indians (Sulbasutras) and the
Babylonians.
Pythagoras
Very little is known about the ancient Greek philosopher and mathematician, Pythagoras
of Samos. He is known to have formed a cult like group devoted to the study of
numbersii. He valued secrecy, and shared these beliefs with his followers known as
Pythagoreans, which is why much is unknown about him. The beliefs of his Pythagorean
society included:
• All things are numbers. Mathematics is the basis for everything, and geometry
is the highest form of mathematical studies. The physical world can be
understood through mathematics.
• The soul resides in the brain, and is immortal. It moves from one being to
another, sometimes from a human into an animal, through a series of
reincarnations called transmigration until it becomes pure. Pythagoras
believed that both mathematics and music could purify.
• Numbers have personalities, characteristics, strengths and weaknesses.
• The world depends upon the interaction of opposites, such as male and
female, lightness and darkness, warm and cold, dry and moist, light and
heavy, fast and slow.
• Certain symbols have a mystical significance.
• All members of the society should observe strict loyalty and secrecy.iii
It should also be noted that some historians doubt whether or not Pythagoras ever proved
the theorem, and there is uncertainty about whether it can be attributed to him directly or
one of his followersiv.
The Babylonians
From about 4000 BC to 500 BC the Babylonians inhabited Mesopotamia, Lying between
the Tigris and the Euphrates, Mesopotamia means” between the rivers” in Greekv. Today
it would lie mostly in Iraq, with some parts of Turkey, Syria, and Iran. The Babylonians
invaded the area first inhabited by the Sumerians. The Sumerians were quite advanced
with irrigation, postal service, and a writing system along with many other advancements.
When the Babylonians tool control, they adopted the Sumerian writing system called
cuneiform. Cuneiform writing consists of symbols drawn into wet clay that is then
allowed to dry into hard tabletsvi. These tablets give us much of what we know about the
Babylonians. Babylonians are well known for their contributions to the field of
mathematics and geometry. They had a place value system based on the number 60, and
were the first to divide the day into 24 hours (60 minutes per hour, 60 seconds per
minute). They used complex calculations, figured out squares up to 59, and were aware
of the properties of Pythagoreans theorem long before he was born.vii One of the
Babylonian tablets reads:
“4 is the length and 5 the diagonal. What is the breadth?
Its size is not known.
4 times 4 is 16.
5 times 5 is 25.
You take 16 from 25 and there remains 9.
What times what shall I take in order to get 9?
3 times 3 is 9.
3 is the breadth.”viii
Clearly this follows the form of Pythagoreans later theorem. Today a geometry student
would work this out in much the same way:
The Babylonians also used these notions in more complex ways. Another tablet shows
how a diagonal of a square by multiplying its side by the square root of two. Yet another
lists Pythagorean triples and another uses these ideas to determine the radius of a circle
with an isosceles triangle inside of it. ix
The Sulbasutras
The Sulbasutras are not a group of people, but a group of texts. They were written by the
Vedic people who lived in India from 1500 BC – 200 BC, and are a set of rules for how
to make altarsx. A very important part of their Vedic religion were sacrificial ceremonies,
of food or sometimes animals, that took place at intricate altars. According to their beliefs
more accurate the measurements led to better ceremonies. They gave rules for creating
geometric shapes including specifics such as making squares with equal area to
rectangles and circles of equal area to squares. xi Several of the authors of the Sulbasutras
are better known including the Baudhayana and the Katyayana. Both of these authors
include information related to Pythagorean theorem in their writings. The Baudhayana
wrote:
“The rope which is stretched across the diagonal of a square produces an area
double the size of the original square.”
A more general description was given by The Katakana. He said:
“The rope which is stretched along the length of the diagonal of a rectangle
produces an area which the vertical and horizontal sides make together.”xii
This very closely resembles Pythagoras’ theorem, if the rope is the hypotenuse of a right
triangle, as is illustrated in the following figure:
Figure from: The Indian Sulbasutras, O’Conner and Robertson @
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html
The Sulbasutras also include Pythagorean triples, and more complex constructions such
as making a square that has the same area as two different squares, their areas given,
using the Pythagorean theorem, as well as using it for the aforementioned squares equal
to areas of rectangles and circlesxiii.
Proofs
There are literally hundreds of ways to prove the Pythagorean theorem using different
topics and types of mathematics and geometry. The following two are more accessible
and common.
Proof 1:
This first proof is based on Euclid’s proof of the Pythagorean theorem found at
http://www.cut-the-knot.org/pythagoras/index.shtml.
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.
This proof illustrates why the Pythagorean theorem works. I like it because it utilizes the
triangle concgurence ideas which are a part of the 7th grade content standards. It is a little
more advanced and abstract or hard to visualize. The second proof is a more visual one,
and is based more in algebra, which might make it more comfortable for students to
interact with.
Proof 2
This proof is worked out using Mathematics for Elementary Teachers workbook Activity
11F.
Proof: a + b = c
Consider the following right triangle.
Set 1:
Make 8 identical triangles
And three squares with sides
The lengths of the sides of
The triangle a, b, and c
Set 2:
Notice how each set of shapes fits into a square with sides a+b, to simplify I will call the
area of the triangle with legs a and b and hypotenuse c, x.
Set 1:
a + b + 4x
Set 2:
c + 4x
It is easy to see that the first square with area a+b is made up of four of our original
triangles plus a square with area a, and a square with area b. The second square with area
a+b is made up of again four triangles and the square with area c. Therefore, the area of a
and b must be equal to the area of c.
Another way of writing this is using algebra:
Simplifying these equations gives the Pythagorean theorem
Set 1: Set 2:
a + b + 4x = a+b
c + 4x = a+b
a + b + 4x = c + 4x
a+b=c
This proof would be more possible for 8th grade students to use because it has a lot to do
with area which is something they are comfortable with, as well as congruent triangles,
rotating and flipping, simple algebra, and could be a hands on experience (have students
cut out the shapes and play with how they fit together).
Development:
While the grade level content expectation
G1.2.3 – Know a proof of the Pythagorean Theorem and use the Pythagorean Theorem
and its converse to solve multistep problems
is a high school objective, concepts related to the understanding of the Pythagorean
theorem can be traced back to kindergarten. In the age of accountability it is important to
be able to direct what we teach back to benchmarks and expectations. The following
GLCEs are related to the Pythagorean theorem. Depending on the proof a high school
student is seeking to understand, different math skills are necessary. I will include
benchmarks with build up to the understanding of one or the other of the proofs listed
above, as well as the theorem generally.
Building Shape Concepts
In order to later understand the Pythagorean theorem, students must have a strong
understanding of shapes and their properties, specifically of triangles and squares. This
starts in Kindergarten with shape recognition and continues through grade 4 when
students are able to describe what type of a triangle a given illustration depicts.
Specifically, being able to recognize a right triangle is crucial in applying the
Pythagorean theorem.
G.GS.00.03 Create, describe, and extend simple geometric patterns.
G.SR.01.03 Create and describe patterns, such as repeating patterns and growing
patterns using number, shape, and size.
G.GS.02.01 Identify, describe, and compare familiar two-dimensional and threedimensional shapes, such as triangles, rectangles, squares, circles, semi-circles, spheres,
and rectangular prisms.
G.GS.02.02 Explore and predict the results of putting together and taking apart twodimensional and three-dimensional shapes.
G.GS.04.02 Identify basic geometric shapes including isosceles, equilateral, and right
triangles, and use their properties to solve problems.
Using Shape Concepts
Having this basic understanding of shapes and their properties, leads students to be able
to play with them and apply what they know to more complicated problems. This is
moving students in the direction needed to use and prove the Pythagorean theorem.
G G.SR.03.05 Compose and decompose triangles and rectangles to form other familiar
two-dimensional shapes, e.g., form a rectangle using two congruent right triangles, or
decompose a parallelogram into a rectangle and two right triangles.
G.GS.05.07 Find unknown angles and sides using the properties of: triangles, including
right, isosceles, and equilateral triangles; parallelograms, including rectangles and
rhombuses; and trapezoids.
M.TE.04.10 Identify right angles and compare angles to right angles.
Towards Understanding Congruency
Not implicit in the Pythagorean theorem itself, the idea of triangle congruence is central
to some proofs of the theorem including the ones shown above. Understanding that
shapes can be moved around and still be the same, and that triangles with similar
properties except maybe for location and orientation are congruent is the first step. Then
students must be able to recognize the criteria for knowing if two triangles are congruent.
Also, understanding the Pythagorean theorem might help a student to know if triangles
are congruent by giving them the information necessary to use one of the triangle
congruence rules.
G.TR.02.06 Recognize that shapes that have been slid, turned, or flipped are the same
shape, e.g., a square rotated 45° is still a square
G.TR.04.05 Recognize rigid motion transformations (flips, slides, turns) of a twodimensional object.
G.TR.06.03 Understand the basic rigid motions in the plane (reflections, rotations,
translations), relate these to congruence, and apply them to solve problems.
G.TR.07.05 Show that two triangles are similar using the criteria: corresponding angles
are congruent (AAA similarity); the ratios of two pairs of corresponding sides are equal
and the included angles are congruent (SAS similarity); ratios of all pairs of
corresponding sides are equal (SSS similarity); use these criteria to solve problems and
to justify arguments.
Necessary Number and Algebra Basics
In order to use the Pythagorean theorem, students must have an understanding of how to
solve for a variable. If given a and c for example, students need to be able to solve for b
in a + b = c. That knowledge is then transferred to the Pythagorean theorem when the
squares are present. Also, in order to be able to work through proofs, it is necessary for
students to understand how to simplify equations and set up problems to solve them.
Before using algebra, students must have strong basic arithmetic skills.
N.MR.03.09 Use multiplication and division fact families to understand the inverse
relationship of these two operations, e.g., because 3 x 8 = 24, we know that 24 ÷ 8 = 3
or 24 ÷ 3 = 8; express a multiplication statement as an equivalent division statement.
A.FO.06.11 Relate simple linear equations with integer coefficients, e.g., 3x = 8 or
x
+ 5 = 10, to particular contexts and solve.
A.FO.06.12 Understand that adding or subtracting the same number to both sides of an
equation creates a new equation that has the same solution.
Using and Understanding Area
A thorough understanding of area is important in using the Pythagorean theorem.
Without this knowledge, students will not have the skills necessary to understand what
the squares mean and how the numbers in the theorem are related. They will not be able
to follow proofs of the theorem, which draw on the area of squares. Understanding how
to write the area of a square in square units will help them understand squares more
generally, as well. Even before doing problems with area, students must be confident
using multiplication and how it can be used to solve problems.
N.MR.02.13 Understand multiplication as the result of counting the total number of
objects in a set of equal groups, e.g., 3 x 5 gives the number of objects in 3 groups of 5
objects, or 3 x 5 = 5 + 5 + 5 = 15.
N.MR.02.13 Understand multiplication as the result of counting the total number of
objects in a set of equal groups, e.g., 3 x 5 gives the number of objects in 3 groups of 5
objects, or 3 x 5 = 5 + 5 + 5 = 15.
M.UN.03.05 Know the definition of area and perimeter and calculate the perimeter of a
square and rectangle given whole number side lengths.
M.UN.03.06 Use square units in calculating area by covering the region and counting
the number of square units.
M.UN.03.07 Distinguish between units of length and area and choose a unit appropriate
in the context.
M.TE.02.04 Find the area of a rectangle with whole number side lengths by covering
with unit squares and counting, or by using a grid of unit squares; write the area as a
product.
M.TE.04.07 Find one dimension of a rectangle given the other dimension and its
perimeter or area.
M.TE.04.08 Find the side of a square given its perimeter or area.
M.PS.04.09 Solve contextual problems about perimeter and area of squares and
rectangles in compound shapes.
Which Lead us too
G.GS.08.01 Understand at least one proof of the Pythagorean Theorem; use the
Pythagorean Theorem and its converse to solve applied problems including perimeter,
area, and volume problems.
and beyond…
Activities
Activity 1: Area Extended
(Created by Alison Keller)
Grade Level: 4
Objectives: This activity will focus on several fourth grade standards and how they relate
to the Pythagorean theorem, without ever discussing the actual theorem and what it
means. These standards include:
M.TE.02.04 Find the area of a rectangle with whole number side lengths by covering
with unit squares and counting, or by using a grid of unit squares; write the area as a
product.
M.TE.04.07 Find one dimension of a rectangle given the other dimension and its
perimeter or area.
M.TE.04.08 Find the side of a square given its perimeter or area.
M.PS.04.09 Solve contextual problems about perimeter and area of squares and
rectangles in compound shapes.
In order to reach these objectives students will:
• Review area of rectangles and squares
• Solve more complex problems using area
• Begin to think about the relationships of the sides of a right triangle
• Begin to think about Pythagorean triples and theorem
Materials: worksheet, pencils, partner
Procedure
• Students will work with a partner to complete the attached worksheet
• Teacher will walk through the room checking progress
• After about 15-20 minutes the class will share their findings, and results will be
recorded on a math bulletin board
• This assignment will be extended in the days following to comparing the area of
squares similarly attached to other right triangles. These will additionally be
recorded on the bulletin board.
Rationale: This activity will get students thinking about something they began to learn
about in third grade, area. They will expand their knowledge of calculating the area of
squares and rectangles, and compound shapes. Then it will encourage students to be
thinking about the Pythagorean triple 3,4,5. It will begin to show the relationships of the
Pythagorean theorem without ever actually introducing the theorem or the concept of
squares. Some of the problems may be a challenge, but teacher assistance can help
students through, and more advanced students will appreciate the challenge. Class
discussion will catch up children who had a harder time but introducing them to the ideas
of their classmates. Hopefully, they will see the relationships (leg squared + leg squared
= hypotenuse squared) in this triangle, and in other right triangles.
Activity 2: Pythagorean Theorem Web Exploration
(Using and modifying PBS’s Nova Series on the Proof, Pythagorean Theorem)
Grade Level: 8
Objectives: This activity will work on the 8th grade geometry standard, “understand and
use the Pythagorean Theorem” G.GS.08.01 Understand at least one proof of the
Pythagorean Theorem; use the Pythagorean Theorem and its converse to solve applied
problems including perimeter, area, and volume problems.
In order to meet these objectives, students will,
• Students will see the Pythagorean theorem in action
• Students will practice using the theorem
• They will apply the theorem to interesting problems
• They will use the theorem in relation to area and perimeter of triangles
Materials: computer with Internet access, pencil, paper
Procedure:
• Students will go to the following website:
http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html
• They will work individually, but may consult their neighbors
• As they work they will write down the answers to the following questions
o What is a?
o What is b?
o What is c?
o Fill in these numbers into the Pythagorean theorem and show how it work
• Then they will click “solve real problems”
• They will do the following:
• Baseball
o Draw a diagram of the problem and solve. Please try it before you ask for
a hint.
o What is the area of the triangle you made to solve this?
o Without doing any additional math, how far would the catcher have to
throw the ball from home to second to throw a runner out there?
• Ladders
o Draw a diagram of the problem and solve.
o What is the perimeter of the triangle you made to solve this?
o If someone builds a moat to keep you out that went 30 feet away from the
house, how long would your ladder have to be?
Rationale: This activity gets students using technology, which is something I think is a
necessary part of classroom instruction. It gives an interactive example of the
Pythagorean theorem that will help visual learners as well as kinesthetic learners. It also
gives realistic situations, which make using the Pythagorean theorem more interesting.
Having a chance to use the computer will also make this assignment fun as well as
educational.
Modern Application
On June 5, 2006, a judge in Albany New York threw out a case about whether double
bunking inmates in prison was un-constitutional. The lawyer used the Pythagorean
theorem to calculate distances of inmates from the toilets in their cells, saying that they
were meant for one person, rather than two. The argument was that those in 2 person cells
had to sleep closer to the toilet than those in one person cells “Inmates at 13 maximumsecurity prisons argued that they were more likely to be assaulted, faced higher chances
of catching a disease and suffered harsh living conditions.xiv” While the article did not go
into detail, I imaging the type of work this lawyer did looked something like:
While initially it appears that the Pythagorean theorem is a topic specifically reserved for
high school geometry, it is clear that its foundations begin in early elementary school. An
important part of history, the theorem has been studied for thousands of years, and is still
applicable today, in the “math class world” and in the “real world.”
i
Mathematics for Elementary Teachers, Sybilla Beckman, pg. 488. Pearson Education
Pythagoras of Samos @ http://scienceworld.wolfram.com/biography/Pythagoras.html
iii
Biography of Pythagoras, C. Douglas @ http://www.mathopenref.com/pythagoras.html
iv
Pythagoras, Stanford Encyclopedia of Philosophy @
http://plato.stanford.edu/entries/pythagoras/
v
Mesopotamia, The British Museum @
http://www.mesopotamia.co.uk/geography/home_set.html
vi
An Overview of Babylonian Mathematics, O’Conner and Robertson, 2000. @
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_mathematics.html
vii
An Overview of Babylonian Mathematics, O’Conner and Robertson, 2000. @
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_mathematics.html
viii
Pythagoras’s theorem in Babylonian Mathematics, O’Conner and Robertson @
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_Pythagoras.html
ix
Pythagoras’s theorem in Babylonian Mathematics, O’Conner and Robertson @
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_Pythagoras.html
x
The Indian Sulbasutras, O’Conner and Robertson @
ii
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html
The Indian Sulbasutras, O’Conner and Robertson @
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html
xii
The Indian Sulbasutras, O’Conner and Robertson @
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html
xiii
The Indian Sulbasutras, O’Conner and Robertson @
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html
xiv
Judge Dismisses Double Bunking Lawsuit, M. Johnson, ABC News @
http://abcnews.go.com/US/wireStory?id=2041612
xi