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Math Help - Why does the pythagorean theorem work?

  1. #1
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    Why does the pythagorean theorem work?

    One of my students today asked my why a^2 + b^2 = c^2

    They continued to say why isn't it a + b = c?

    Thanks to the help of all of you here, many times I have been prepared to answer these questions. But today I found myself with no other answer than "That is what the formula tells us."

    As a teacher and former/current student, I HATE THAT ANSWER.

    So I told my student I would find out why and so far you guys here have been a tremendous asset to my classroom and I really appreciate all the time you put in. If anyone can help me out that would be greatly appreciated.
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  2. #2
    Super Member TheChaz's Avatar
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    First, a + b doesn't equal (usually*) c by the triangle inequality.
    For a proof of the PT, you could see Euclid's Elements book I.

    But I prefer http://www.cut-the-knot.org/pythagoras/
    It will blow your mind., especially some of the graphical examples.
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  3. #3
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    This is a very favorite question of mine. It comes up in the out of Africa debates.
    What could be more important to builders that the getting things square?
    Squares make structures the most stable. The Egyptians were the oldest and perhaps best builders of the BCE world. They knew that by creating any 3, 4, 5 triangle a ‘perfect’ square measure was made. In other words, that knew from experience how to construct a builder’s set square. There is absolutely no evidence they ever considered the 3^2+4^4=5^2 relationship. In fact the opposite may be true.
    There is an abundance of evidence that the Babylonians knew of thousands of these triples that accomplished the same result. But that said, there is absolutely no evidence that they understood that if we start with any right triangle that we will get one of those triples.

    That connection was establishes by the Greek religious sect established by Pythagoras in the 5th century BCE. Thus the name Pythagorean Theorem

    This is a case where reality drove the proof of a mathematical result.

    Here is a recap: It was know for a long time that the relation certain triples would give a right triangle. But then it was proved that any right triangle will give a triple that has the relation a^2+b^2=c^2.

    There is a good discussion of this in The Assent of Man by Bronowski.
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  4. #4
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    Google proof using similar triangles

    bjh
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    Quote Originally Posted by bjhopper View Post
    Google proof using similar triangles
    I really doubt that has anything to with this question.
    The question is really philosophical and not mathematical.
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    Note that all the right-angle triangles are of equal lengths \displaystyle a, b, c. You need to note that the big squares are equal in area, so the remaining areas when you remove the right-angle triangles are also equal in area.
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  7. #7
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    Quote Originally Posted by Jman115 View Post
    One of my students today asked my why a^2 + b^2 = c^2

    They continued to say why isn't it a + b = c?

    Thanks to the help of all of you here, many times I have been prepared to answer these questions. But today I found myself with no other answer than "That is what the formula tells us."

    As a teacher and former/current student, I HATE THAT ANSWER.

    So I told my student I would find out why and so far you guys here have been a tremendous asset to my classroom and I really appreciate all the time you put in. If anyone can help me out that would be greatly appreciated.
    It is a characteristic of Euclidean space (it is equivalent to the parallels postulate)

    CB
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    To answer the question "Why doesn't \displaystyle a + b = c?", well think about a right-angle triangle. The shortest distance between any two points is a straight line segment. The hypotenuse ( \displaystyle c) is this shortest distance. The other two sides, since they are not a straight line segment, can not possibly be the shortest distance. Therefore \displaystyle a + b \neq c.
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    You did show them a proof right? If not, that's the first thing you should do. You could also draw an accurate right triangle and demonstrate by measuring that their guess is wrong.

    If they've seen a proof and still ask this, then I'd say that they have a serious gap in their understanding of math and maybe taking some class time to go over the concept of proof would be helpful.
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  10. #10
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    Not to be wise, but the question I asked kind of implies I didn't show them a proof. Not to mention they are 7th graders. 12 years old. Half of them struggle with their multiplication facts.

    Prove it, thank you so much for your responses. I love how you always provide pictures, they work great with my students when trying to explain this stuff.

    I am still looking at the two squares with right triangles in them. I may have a question about that one here in a bit. But I also know it usually takes a little time to sink in with me.
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  11. #11
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    2 Minutes later I have my duh moment. That picture is PERFECT thank you so much. This will work great with the students.
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  12. #12
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    Quote Originally Posted by Jman115 View Post
    Not to be wise, but the question I asked kind of implies I didn't show them a proof. Not to mention they are 7th graders. 12 years old. Half of them struggle with their multiplication facts...
    I don't mind you being wise/fresh, since the reply to which you refer implies that *you* have a serious gap in your understanding! And for another shameless plug of the link I posted, "note" that there are many more pictures!
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  13. #13
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    Quote Originally Posted by TheChaz View Post
    I don't mind you being wise/fresh, since the reply to which you refer implies that *you* have a serious gap in your understanding!
    I taught for a few years. Trust me when I say that a lot of my students left my class with massive misconceptions and downright false beliefs about math. No matter how hard I tried or how many times I explained that math makes sense, I think some of them believe that formulas and theorems were just arbitrarily created a long time ago. One time a junior was surprised that -1/1=1/-1. Does that mean that I have a serious gap in my understanding of negatives? I was just trying to suggest that things which seem completely clear and obvious to teachers can be confusing to students.
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  14. #14
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    Most teachers don't teach why things work. Part of what makes me a good teacher is I do. When I don't know or forgot the answer to a question like this, I ask. I model this for my students.

    Having not been taught this one concept in no way shape or form means I have a significant lack in my understanding of mathematics. I wasn't being wise, or fresh. I was just pointing out it out.

    Prove It gave me a very simple, straight forward answer that also provided me with a great visual for 7th grade minds to wrap their heads around. Your link was nice, but wasn't as well packaged for an explanation to 12 year olds.

    Plato gave me some good history to help illustrate a story for my students as I explained which worked out very nicely!
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  15. #15
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    This animation might also be of some use

    Perigal's Proof of the Pythagorean Theorem

    It basically shows you that no matter how you resize the triangle, by cutting up the squares of the smaller sides correctly, they can be arranged to form the square on the hypotenuse.
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