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1. "The Pythagorean Theorem: A 4,000 Year Old History" by Eli Maor

2. "Hidden Harmonies" By Robert and Ellen Kaplan

3. "Pythagoras, What's Your Angle?" by Julie Ellis

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An image that will be utilized in order to demonstrate how the theorem facilitates the derivation of the equation of a circle.

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Three Examples of Usage of Technology During This Unit

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Three Examples of Usage of Technology During This Unit

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1. After the students first discover the theory on their own (sage on stage vs. guide on the side), the first lesson in the unit will be concluded with a video about Pythagoras, his life, and an overview of his theory. http://bit.ly/1ES3uIj

2. As a mid-unit assessment, on a smart board,  the teacher will begin creating different right triangles using GEOMETER’s SKETCHPAD. Each one will have a measurement missing. The students will use the theorem to solve for the missing sides.

3. The final assessment of the unit lesson will not only include a paper-and-pencil test with regards to the history of Pythagoras and computing sides of triangles and the equations of circles, but will also utilize the InterACTIVE Pythagorean Explorer. http://bit.ly/1vkc6Wu

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Here are 10 more practical uses of the Pythagorean Theorem.

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Real Life Uses of the Pythagorean Theorem

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From architecture and construction to sailing and space flight, the Pythagorean Theorem has wealth of real-life uses, some of which you may already use.

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In case you were interested in more life applications...

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In case you were interested in more life applications...

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Here is a fascinating article about some other applications of the theorem.

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Putting aside the fact that Pythagoras had a struck of luck when he fell from his bed that morning of July of 555 B.C. in breeze Samos Greece. He actually saw polynomials instead of birds. If we could make our kids understand better his horrifying formula, we could save this country to be relegated to a third world mind.
HANDS ON...
Let's assume that you already have seen the following formula and you want to use it in real pulsing timing:

a² +b² =c²
HERE WE GO
You live on a 5th floor on a building near Downtown Los Angeles. Suddenly, your freaking neighbor, who lives on the 4th floor, starts (ignites) a fire. All because he caught his girl chatting with another guy-- familiar?

Not that she was talking to a colleague on FB. Not that we know or were told. But before we know it, he bashes the whole apartment to sheer delight. He leaves her girlfriend unconscious and you happen to see him running downstairs. You become a hero and take things into matter. You decide to drag her out across the hallway -- flames catch up with you both. The only way to stay alive is keep going up into the top floor and roof. You call 911 and they get hold of your emergency.
SPEEDING UP THE STORY
A fire engine gets there in 6 minutes. You are by the top floor window and hope and pray that the telescopic aerial ladder will be able to reach you both.
DO WE NEED MATH?
A fireman does a quick calculation:

We have 5 floors, each floor is 8.5 feet high ( Hurry up Lord!!)

Total height of the target is 8.5x5= 42.5 feet

The closest an Engine can get (sideways) is roughly 7 feet ( Lord please!!)
We don't worry about nothing else and apply the theorem:
(42.5)² + (7)2 = (ladder span needed)2 = L²

The fire-guys will solve this equation in 5 seconds...

1806.25 + 49 = L²

L2 = 1855.25 Then L= 43.072613108563544 or: L=43.1 FEET

The ladder span is 43.1 feet!

YES! The aerial span needed is about 43 feet, and luckily the fire engine mount aerial ladder can be extended up to 50 feet. Our 'finest' can go up, with no problem at all.
SAVED BY THE SMOKING BELL THAT WILL DESTROY ITSELF IN 5 SECONDS!
On the picture above, the unionized brother contractor made an approximate ramp length. What would've happened if he knew our method?

We have the height from the ground to sliding door porch which is 2 feet.

The actual distance from the bottom side of the sliding door to the sidewalk is 23 feet as you see.

Lets call the actual length of the ramp X (in the picture is represented by ??)

x² =23² + 2²

x² = 529 + 4

x2 =533
Solving the square root of 533, we obtain the answer x= 23.086 feet So, the Contractor will be in better shape by knowing the length of the ramp in advance. The Ramp will be 23.086 feet long.
..

You can remember the formula by using mnemonics.

3 2 + 4² = 5 2
3 persons have ducks on their head in one corner
4 persons have more ducks on their head on the other corner
How many persons will be able to match that kind of setting, being caught in the middle of the room with ducks over their head? 5

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Pythagorean Theorem -- Geometry and Applications in Real Life

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Putting aside the fact that Pythagoras had a struck of luck when he fell from his bed that morning of July of 555 B.C. in breeze Samos Greece. He actually saw polynomials instead of birds. If we could make our kids understand better his horrifying formula, we could save this country to be relegated to a third world mind.

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This is the question that students have asked from time immemorial. Check out this article which begins to touch upon how useful the Pythagorean Theorem is in real life, while also simultaneously reviewing the key elements of the theorem and its proof.

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Pythagoras was a Greek philosopher and mathematician. Whether you know it or not, Pythagoras's theorem, named after him, is used by almost everybody in real life experiences.

Uses of Pythagoras' Theorem

You may have heard about Pythagoras's theorem (or the Pythagorean Theorem) in your math class, but what you may fail to realize is that Pythagoras's theorem is used often in real life situations. Gain a better understanding of the concept with these real-world examples.

According to Pythagoras's theorem the sum of the squares of two sides of a right triangle is equal to the square of the hypotenuse. Let one side of the right triangle be a, the other side be b and hypotenuse is given by c. According to Pythagoras's theorem

a ² + b ²= c ²

This is taught in every classroom throughout the world, but what isn't taught is how it can be applied outside of the classroom.

Real Life Applications

Some real life applications to introduce the concept of Pythagoras's theorem to your middle school students are given below:

1) Road Trip: Let's say two friends are meeting at a playground. Mary is already at the park but her friend Bob needs to get there taking the shortest path possible. Bob has two way he can go - he can follow the roads getting to the park - first heading south 3 miles, then heading west four miles. The total distance covered following the roads will be 7 miles. The other way he can get there is by cutting through some open fields and walk directly to the park. If we apply Pythagoras's theorem to calculate the distance you will get:

(3) 2 + (4) ² =

9 + 16 = C ²

√25 = C

5 Miles. = C

Walking through the field will be 2 miles shorter than walking along the roads. .

2) Painting on a Wall: Painters use ladders to paint on high buildings and often use the help of Pythagoras' theorem to complete their work. The painter needs to determine how tall a ladder needs to be in order to safely place the base away from the wall so it won't tip over. In this case the ladder itself will be the hypotenuse. Take for example a painter who has to paint a wall which is about 3 m high. The painter has to put the base of the ladder 2 m away from the wall to ensure it won't tip. What will be the length of the ladder required by the painter to complete his work? You can calculate it using Pythagoras' theorem:

(5) 2 + (2) ² =

25 + 4 = C ²

√100 = C

5.3 m. = C

Thus, the painter will need a ladder about 5 meters high.

3) Buying a Suitcase: Mr. Harry wants to purchase a suitcase. The shopkeeper tells Mr. Harry that he has a 30 inch of suitcase available at present and the height of the suitcase is 18 inches. Calculate the actual length of the suitcase for Mr. Harry using Pythagoras' theorem. It is calculated this way:

(18) 2 + (b) ² = (30) ²

324 + b ² = 900

B ² = 900 - 324

b= √576

= 24 inches

4) What Size TV Should You Buy? Mr. James saw an advertisement of a T.V.in the newspaper where it is mentioned that the T.V. is 16 inches high and 14 inches wide. Calculate the diagonal length of its screen for Mr. James. By using Pythagoras' theorem it can be calculated as:

(16) 2 + (14) ² =

256 + 196 = C ²

√452 = C

21 inches approx. = C

5) Finding the Right Sized Computer: Mary wants to get a computer monitor for her desk which can hold a 22 inch monitor. She has found a monitor 16 inches wide and 10 inches high. Will the computer fit into Mary's cabin? Use Pythagoras' theorem to find out:

(16) 2 + (10) ² =

256 + 100 = C ²

√356 = C

18 inches approx. = C
Practice Ideas

Now write your own problem based on a potential real life situation.

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Uses of Pythagoras's Theorem in Real Life Scenarios for Middle School Students

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You may have heard about Pythagoras's theorem (or the Pythagorean theorem) in your math class, but how well do you understand it? Pythagoras's theorem is actually used quite often in real life situations. Gain a better understanding of the concept with these five examples.

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The Pythagorean Theorem also helps us derive the equation of a circle. Read the attached presentation and see how this is even possible.

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Derive the equation of a circle: using the Pythagorean Theorem

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In this lesson you will learn how to derive the equation of a circle by using the Pythagorean Theorem.

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Here is a worksheet to practice finding the distance between two points on a graph using Pythagoras's famous theorem.

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Math-Aids.Com | Dynamically Created Math Worksheets

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This page is for viewing the dynamically created Math Worksheets from the different math generator pages.

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The Pythagorean Theorem isn't only something  associated with a, b, and c. We can also use it to help us find the distance between points on a graph. We will discuss this in detail in class, but here is a video to give you a head start.

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Using The Distance Formula Or Pythagorean Theorem To Find The Distance Between Two Points

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This tutorial examines how to find the distance between two points that are shown on the coordinate plane by drawing a right triangle and using the Pythagorean Theorem to solve. Also see how to use the distance formula when only given the ordered pairs of two points.

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There have been various proofs found for the Pythagorean Theorem over the centuries. One was even found by a United States President!

We will discuss four of these proofs, found on this website, in depth. Get a head start by clicking on the attachment and checking them out!

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Pythagorean Theorem

by

Angie Head

This essay was inspired by a class that I am taking this quarter. The class is the History of Mathematics. In this class, we are learning how to include the history of mathematics in teaching a mathematics. One way to include the history of mathematics in your classroom is to incorporate ancient mathematics problems in your instruction. Another way is to introduce a new topic with some history of the topic. Hopefully, this essay will give you some ideas of how to include the history of the Pythagorean Theorem in the teaching and learning of it.

We have been discussing different topics that were developed in ancient civilizations. The Pythagorean Theorem is one of these topics. This theorem is one of the earliest know theorems to ancient civilizations. It was named after Pythagoras, a Greek mathematician and philosopher. The theorem bears his name although we have evidence that the Babylonians knew this relationship some 1000 years earlier. Plimpton 322, a Babylonian mathematical tablet dated back to 1900 B.C., contains a table of Pythagorean triples. The Chou-pei, an ancient Chinese text, also gives us evidence that the Chinese knew about the Pythagorean theorem many years before Pythagoras or one of his colleagues in the Pythagorean society discovered and proved it. This is the reason why the theorem is named after Pythagoras.

Pythagoras lived in the sixth or fifth century B.C. He founded the Pythagorean School in Crotona. This school was an academy for the study of mathematics, philosophy, and natural science. The Pythagorean School was more than a school; it was "a closely knit brotherhood with secret rites and observances" (Eves 75). Because of this, the school was destroyed by democratic forces of Italy. Although the brotherhood was scattered, it continued to exist for two more centuries. Pythagoras and his colleagues are credited with many contributions to mathematics.

The following is an investigation of how the Pythagorean theorem has been proved over the years.

Pythagorean Theorem

The theorem states that:

"The square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs" (Eves 80-81).

This theorem is talking about the area of the squares that are built on each side of the right triangle.

Accordingly, we obtain the following areas for the squares, where the green and blue squares are on the legs of the right triangle and the red square is on the hypotenuse.

area of the green square is
area of the blue square is
area of the red square is

From our theorem, we have the following relationship:

area of green square + area of blue square = area of red square or

As I stated earlier, this theorem was named after Pythagoras because he was the first to prove it. He probably used a dissection type of proof similar to the following in proving this theorem.

Pythagoras' Proof

"Let a, b, c denote the legs and the hypotenuse of the given right triangle, and consider the two squares in the accompanying figure, each having a+b as its side. The first square is dissected into six pieces-namely, the two squares on the legs and four right triangles congruent to the given triangle. The second square is dissected into five pieces-namely, the square on the hypotenuse and four right triangles congruent to the given triangle. By subtracting equals from equals, it now follows that the square on the hypotenuse is equal to the sum of the squares on the legs" (Eves 81).

Consider the following figure.

The area of the first square is given by (a+b)^2 or 4(1/2ab)+ a^2 + b^2.
The area of the second square is given by (a+b)^2 or 4(1/2ab) + c^2.
Since the squares have equal areas we can set them equal to another and subtract equals. The case (a+b)^2=(a+b)^2 is not interesting. Let's do the other case.
4(1/2ab) + a^2 + b^2 = 4(1/2ab)+ c^2
Subtracting equals from both sides we have

concluding Pythagoras' proof.

Over the years there have been many mathematicians and non-mathematicians to give various proofs of the Pythagorean Theorem. Following are proofs from Bhaskara and one of our former presidents, President James Garfield. I have chosen these proofs because any of them would be appropriate to use in any classroom.

Bhaskara's First Proof

Bhaskara's proof is also a dissection proof. It is similar to the proof provided by Pythagoras. Bhaskara was born in India. He was one of the most important Hindu mathematicians of the second century AD. He used the following diagrams in proving the Pythagorean Theorem.

In the above diagrams, the blue triangles are all congruent and the yellow squares are congruent. First we need to find the area of the big square two different ways. First let's find the area using the area formula for a square.
Thus, A=c^2.
Now, lets find the area by finding the area of each of the components and then sum the areas.
Area of the blue triangles = 4(1/2)ab
Area of the yellow square = (b-a)^2
Area of the big square = 4(1/2)ab + (b-a)^2
= 2ab + b^2 - 2ab + a^2
= b^2 + a^2

Since, the square has the same area no matter how you find it
A = c^2 = a^2 + b^2,
concluding the proof.

Bhaskara's Second Proof of the Pythagorean Theorem

In this proof, Bhaskara began with a right triangle and then he drew an altitude on the hypotenuse. From here, he used the properties of similarity to prove the theorem.

Now prove that triangles ABC and CBE are similar.
It follows from the AA postulate that triangle ABC is similar to triangle CBE, since angle B is congruent to angle B and angle C is congruent to angle E. Thus, since internal ratios are equal s/a=a/c.
Multiplying both sides by ac we get
sc=a^2.

Now show that triangles ABC and ACE are similar.
As before, it follows from the AA postulate that these two triangles are similar. Angle A is congruent to angle A and angle C is congruent to angle E. Thus, r/b=b/c. Multiplying both sides by bc we get
rc=b^2.

Now when we add the two results we get
sc + rc = a^2 + b^2.
c(s+r) = a^2 + b^2
c^2 = a^2 + b^2,
concluding the proof of the Pythagorean Theorem.

Garfield's Proof

The twentieth president of the United States gave the following proof to the Pythagorean Theorem. He discovered this proof five years before he become President. He hit upon this proof in 1876 during a mathematics discussion with some of the members of Congress. It was later published in theNew England Journal of Education.. The proof depends on calculating the area of a right trapezoid two different ways. The first way is by using the area formula of a trapezoid and the second is by summing up the areas of the three right triangles that can be constructed in the trapezoid. He used the following trapezoid in developing his proof.

First, we need to find the area of the trapezoid by using the area formula of the trapezoid.
A=(1/2)h(b1+b2) area of a trapezoid

In the above diagram, h=a+b, b1=a, and b2=b.

A=(1/2)(a+b)(a+b)
=(1/2)(a^2+2ab+b^2).

Now, let's find the area of the trapezoid by summing the area of the three right triangles.
The area of the yellow triangle is
A=1/2(ba).

The area of the red triangle is
A=1/2(c^2).

The area of the blue triangle is
A= 1/2(ab).

The sum of the area of the triangles is
1/2(ba) + 1/2(c^2) + 1/2(ab) = 1/2(ba + c^2 + ab) = 1/2(2ab + c^2).

Since, this area is equal to the area of the trapezoid we have the following relation:
(1/2)(a^2 + 2ab + b^2) = (1/2)(2ab + c^2).

Multiplying both sides by 2 and subtracting 2ab from both sides we get

concluding the proof.

Return to Angela Head's EMT 668 Page

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by This essay was inspired by a class that I am taking this quarter. The class is the History of Mathematics. In this class, we are learning how to include the history of mathematics in teaching a mathematics. One way to include the history of mathematics in your classroom is to incorporate ancient mathematics problems in your instruction.

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This video explains one of the simplest proofs for the Pythagorean Theorem.

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The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs. This video shows a geometric proof based on rearranging triangles in a rectangle. If you like my videos, you can support me at Patreon: http://www.patreon.com/mindyourdecisions Connect on social media.

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4 Common Core Standards that Lessons in this THEMATIC Unit will Address

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4 Common Core Standards that Lessons in this THEMATIC Unit will Address

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1. CCSS.Math.Content.8.G.B.6., Geometry » Understand and apply the Pythagorean Theorem. Explain a proof of the Pythagorean Theorem and its converse.

2. CCSS.Math.Content.8.G.B.7.Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

3. CCSS.Math.Content.8.G.B.8.,Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

4. G. GPA. 1 >>Derive the equation of a circle of given center and radius using the Pythagorean Theorem.

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1.  Identify the parts of a triangle as leg, leg, and hypotenuse

2. Evaluate the Pythagorean theorem and its proofs

3. Analyze a triangle and utilize the Pythagorean theorem to find the value of any side

4. Compare and contrast the proofs that have been developed over the years

5. Use the Pythagorean theorem to discover distances between points on a graph

6. Derive the equation of a circle using the Pythagorean Theorem

7. Describe who Pythagoras was and what mathematical principles he discovered, along with the school that he created

8. Discover and discuss when the Pythagorean theorem is used in real life

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1. Who was Pythagoras and what is the Pythagorean theorem?
2. What are the proofs for the Pythagorean theorem?

3.  In what ways was the Pythagorean theorem an
important addition to the field of mathematics?
4. How is the Pythagorean Theorem applicable in everyday
life?

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One of the many proofs.

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...and then Egypt

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First Greece....

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Who was Pythagoras; When and Where did he live; What did he (and his school) discover; Why are his theorems so famous?

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569 B.C.E. - 500 B.C.E.

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a2 + b2 = c2 - The Pythagorean Theorem!

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Pythagoras. Enough Said. :)