# The Pythagorean Theorem: An Exploration

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Hi! Today I?m gonna share with you an activity that you can do with your students that can lead them to discovering the Pythagorean Theorem. This is my longest video so far and so I?ve split it up into three sections.

The first section is just an activity that students can do where they?re asked to find as many different sized squares they can using dot paper. Now, by using dot paper what I mean is that the corners of the squares all have to be on the dots that are on the paper. And this is very similar to another activity they may have done that I did in another video where they?re finding areas of different shapes on dot paper.

The second part of the video is the proof that Pythagoras actually came up with for the Pythagorean Theorem. Now, the Pythagorean Theorem is named after Pythagoras but he didn?t invent it. He didn?t discover it. It was known way before he came along and even the proof that?s credited to him, which is the one that I?ll share with you today, is probably not his own invention. It was probably done before him. But he got credit for it so we still call it the Pythagorean Theorem today.

In the third part of the video, I?ll share with you a natural way that students may come across the formula for the Pythagorean Theorem through the numeric work they do through the activity. So I hope you enjoy.

So, the challenge before students is to come up with as many different sized squares as they can using the dots as vertices of the square. So, at first they might come up with a whole bunch of them that may be very obvious, and they?ll come up with for example this square which has area one. And then they can go a little bigger and come up with square of area of four. And then nine, sixteen, and so on. So, we might want to put a limit on the size of squares that they can find. Maybe it?s find all the square up to area one hundred that you can find, or something like that. Just so that they don?t keep making these squares forever in this orientation. So, when students come and say we found them all we can say, ?Nope, you haven?t. Keep looking.? And that?s where the puzzle really starts and the fun begins.

So, students might say, ?Well, I don?t know what else to do.? And that?ll they might mess around a little bit and come up with something like this. This square that is built on this diagonal here. Now, there?s lots of different ways to find the area of this square. And one common way to that would be to divide this up into smaller triangles. And so they?ve done this work before they see that this triangle right here is actually half of the square and they have four halves, so that makes this a square of area two, which is one they did not have before. And then we can say, well keep going. How many more can you find? And so students might keep working and come up with something like this one. Now to find the area of this square, it?s not as obvious of how to divide it up event though they certainly could divide it up. But if they?ve done the area of that paper they?re gonna have this other technique where they build a rectangle around the object and then subtract away smaller triangles. So, if they do that they?re gonna end up building what?s gonna be a square which is a special type of rectangle after all. And they?ll calculate the area of this whole square which they?ll see as nine. And then they?ll have to find the area of these triangles. And as they do this more and more, they?re gonna notice that rather than before where the areas of the triangles were all kind of random, here they?re all going to have the same areas. It?s very symmetrical. So here we have a total of nine square units. Subtracting away these four triangles which each have one square unit, so nine minus four gives us the area of five for our square.

So, students may continue and when they come up with something like this where it?s a very long length, it might not be obvious where to put the other side to make it perfectly rectangular. They may mess around with that a little bit but if they keep repeating this over and over again, they?re going to notice something that?s very awesome and very important. And it?s actually related to algebra. And that this that if we think of this diagonal as having us go down a certain amount and over so here we?ve gone down five over two that defines how much slantiness or the slope of this line. Now we want to rotate this so its on a ninety degree angle. So if we just turned this triangle ninety degrees we end up with this triangle over here and so what we see is that instead of going down five we are now going over five and instead of going across two we are now going up two. And this is the whole idea behind perpendicular lines. That perpendicular lines have negative reciprocal slopes. The reciprocal just meaning that the rise and the run if you will have changed orientations and that?s because we?ve rotated this triangle ninety degrees and one?s gonna be negative and one?s gonna be positive because they?re facing opposite directions. So, this is a real nice indirect preparation for the idea of perpendicular lines and slope.

So, if we keep going around we?ll rotate our triangle again and make another one here and then we can complete our square and just for fun we?ll just complete the larger square around it. And so students can find the are of this entire square and subtract the triangles once again and end up with an area of twenty-nine square units. And so students can keep going and going and see how many different squares they can find and calculating their areas. And what they?re gonna be seeing is they?re gonna be doing the same procedure all over and over again. And that?s when we can start going into something more formal if students don?t discover it for themselves.

We?re going to build a model of what we were previously doing on the dot paper. And rather than starting with the square, we?re going to start with the triangle that we built our square on. We can do that many different ways and we could use a protractor to measure the ninety degree angle to build that. We could use a compass as well. And we could do the construction that way. Or we can even make a right triangle by using the one that is already on a piece of paper, which is what I?m gonna do because I need all the help I can get with arts and crafts.

And so here?s our perfect ninety degree angle and there?s our triangle.

Now what we?re going to do is we?re going to build the model that we saw on the dot paper. So we?re going to make four copies of our triangle and trace them on a new piece of paper. And by doing so we?ve created our square. Then we?re gonna color in each of the four triangles. This is because we are gonna be cutting these out later and we wanna distinguish those triangles from the square with a different color.

Now that we have that we?re gonna label this. Previously, we used numbers for each sides. But now we?re gonna use letters to represent any amount. So, we?re gonna call the long side of our right triangle A, were gonna call the short side B and our hypotenuse we?re going to call C. So, before we were choosing what A and B were and we never chose C. C is just kinda there and we don?t even know what C?s length actually is but that really doesn?t matter. Now as I label these we?re gonna label all four triangles on the inside of the triangle because we?re gonna cut these out and rearrange these and for that reason we are going to do all four triangles with R, A, B, and C. Now that we?ve done that the square that we wanted to calculate the are of in the center is built upon the side length of C. So now we can actually call that square C-squared and that?ll help us distinguish it from the other squares that we may have like the large square of the whole picture for example.

Now that we?ve done that we?re going to cut out that large square and were going to make a copy of it and we are going to do that by tracing that on another sheet of white paper. Once we have that done we?re going to cut out the four triangles and build the second model that looks exactly like the first one. So, now that we have our copy here we see that this C-square here is the exact same area as this square is over here. And what?s interesting about this square here, and the reason why we took these pieces and we copied them here is that if I take this triangle for example here and put it something crazy like this, I now have this really strange looking white area here. But since these triangles are still inside the square this white area is still the same as C-squared even though I don?t know how to calculate it directly, logically the area hasn?t changed. It?s still the same size. So what that tells us is that I can arrange anyway that we would like to. And here is how I would like to do this. So, I?m gonna take this piece and slide it up here. I?m gonna take this one and slide it here and then slide this one over here. And now this area that was this one giant square that was equal to C-squared is now arranged into two smaller areas and these happen to be squares as well. How do we know that they?re squares? Well the length of this square is A and the length of this side of this square is A therefore this is a square that is equal to A-squared. So whatever number we chose for this side, this is the square of that number. And down here we also have a square that?s length B by B so that makes this equal to B-squared. So we took that square that was in this form and we just rearranged it to look like these two squares here. So that tells us that this square plus this square together is equal in area to this square just by this rearrangement method. And what?s very cool is that we can see that this is our original triangle right here is that we wanted to find if we ignore everything else we just wanted to find the are of this square that was built on this diagonal here. And we can do that instead of doing anything else by subtracting a bunch of triangles we can simply find this square here and this square here and add their areas together to find this area over here. Which is what we wanted.

Something else that?s pretty awesome and the reason why I arranged it this particular way is that this should look familiar to students if they?ve come from Montessori because this is just a model of the binomial square. Where we have this length A plus B and this is A plus B squared with the A-squared, the B-squared, and the two rectangles obey B.

So as students start seeing that they?re doing the same thing over and over again, they?re finding the area of the large square and their subtracting away four triangles they might start to come to something like this. So for example in this one where they had the right triangle lengths of five and two, they might say okay the square, the big square is going to be length five plus two, and we always square that. So would give us forty-nine for the whole square. Then we are going to subtract away four triangles and each triangle?s area is going the be one-half the base which is two or five depending on how you look at it, five times two. So we have seven squared which is our large square minus two rectangles of ten and we get those if we put those triangles together to make two rectangles so altogether we have an area of forty-nine minus the twenty from the rectangles which gives us an are of twenty-nine square units.

But if we see students doing that type of work, we can say alright whatever length you choose we?re gonna call one side A and the other side B. and so the length of your square is going to be A plus B. then you?re gonna subtract four triangles. And the areas of those are one-half times A times B. So in reality what we have is A plus B squared minus two AB?s. Now this A plus B squared students may have seen before and if they come from Montessori elementary they would have seen this. And this represents A plus B squared right here where we have four parts. We have an A-squared, because this length here is A this length here is B so this is A plus B, so this is A-squared and we have to of these rectangles here and this is A by B, so this is AB plus another AB is two AB plus this B-squared in the corner minus two AB. And now we have positive two AB and minus two AB so that just gives us A-squared plus B-squared. So if we look back on our numerical example we had side lengths of five and two for our right triangle well if we just square five that gives us twenty five and if we square the two that gives us four and there?s our twenty-nine. So that?s a much faster way of going through all this because we understand that this process is gonna happen every single time. And that?s how we develop the formula. So what students have done through this is they?ve discovered that the square that?s built on the hypotenuse of that triangle is equal to the sums of the squares that are built on the two legs. They?ve discovered the Pythagorean Theorem.

So that?s it. That?s the Pythagorean Theorem. Now, the second way where the students were doing that numerical exploration and you could see that they?re kind of developing a formula. If we observe that then we can definitely go down that path with them and maybe ask them some questions so that they can discover that formula for themselves with the A plus B squared. However, a lot of students might not quite be there and so showing them the geometric interpretation, the one that Pythagoras did, that might be very intriguing for those students. And I think no matter what all the students enjoy seeing that. You can?t see a proof too many times and seeing it in different ways just add depth and richness to the work.

So, I hope you enjoyed the video and keep exploring the Pythagorean Theorem.

The first section is just an activity that students can do where they?re asked to find as many different sized squares they can using dot paper. Now, by using dot paper what I mean is that the corners of the squares all have to be on the dots that are on the paper. And this is very similar to another activity they may have done that I did in another video where they?re finding areas of different shapes on dot paper.

The second part of the video is the proof that Pythagoras actually came up with for the Pythagorean Theorem. Now, the Pythagorean Theorem is named after Pythagoras but he didn?t invent it. He didn?t discover it. It was known way before he came along and even the proof that?s credited to him, which is the one that I?ll share with you today, is probably not his own invention. It was probably done before him. But he got credit for it so we still call it the Pythagorean Theorem today.

In the third part of the video, I?ll share with you a natural way that students may come across the formula for the Pythagorean Theorem through the numeric work they do through the activity. So I hope you enjoy.

So, the challenge before students is to come up with as many different sized squares as they can using the dots as vertices of the square. So, at first they might come up with a whole bunch of them that may be very obvious, and they?ll come up with for example this square which has area one. And then they can go a little bigger and come up with square of area of four. And then nine, sixteen, and so on. So, we might want to put a limit on the size of squares that they can find. Maybe it?s find all the square up to area one hundred that you can find, or something like that. Just so that they don?t keep making these squares forever in this orientation. So, when students come and say we found them all we can say, ?Nope, you haven?t. Keep looking.? And that?s where the puzzle really starts and the fun begins.

So, students might say, ?Well, I don?t know what else to do.? And that?ll they might mess around a little bit and come up with something like this. This square that is built on this diagonal here. Now, there?s lots of different ways to find the area of this square. And one common way to that would be to divide this up into smaller triangles. And so they?ve done this work before they see that this triangle right here is actually half of the square and they have four halves, so that makes this a square of area two, which is one they did not have before. And then we can say, well keep going. How many more can you find? And so students might keep working and come up with something like this one. Now to find the area of this square, it?s not as obvious of how to divide it up event though they certainly could divide it up. But if they?ve done the area of that paper they?re gonna have this other technique where they build a rectangle around the object and then subtract away smaller triangles. So, if they do that they?re gonna end up building what?s gonna be a square which is a special type of rectangle after all. And they?ll calculate the area of this whole square which they?ll see as nine. And then they?ll have to find the area of these triangles. And as they do this more and more, they?re gonna notice that rather than before where the areas of the triangles were all kind of random, here they?re all going to have the same areas. It?s very symmetrical. So here we have a total of nine square units. Subtracting away these four triangles which each have one square unit, so nine minus four gives us the area of five for our square.

So, students may continue and when they come up with something like this where it?s a very long length, it might not be obvious where to put the other side to make it perfectly rectangular. They may mess around with that a little bit but if they keep repeating this over and over again, they?re going to notice something that?s very awesome and very important. And it?s actually related to algebra. And that this that if we think of this diagonal as having us go down a certain amount and over so here we?ve gone down five over two that defines how much slantiness or the slope of this line. Now we want to rotate this so its on a ninety degree angle. So if we just turned this triangle ninety degrees we end up with this triangle over here and so what we see is that instead of going down five we are now going over five and instead of going across two we are now going up two. And this is the whole idea behind perpendicular lines. That perpendicular lines have negative reciprocal slopes. The reciprocal just meaning that the rise and the run if you will have changed orientations and that?s because we?ve rotated this triangle ninety degrees and one?s gonna be negative and one?s gonna be positive because they?re facing opposite directions. So, this is a real nice indirect preparation for the idea of perpendicular lines and slope.

So, if we keep going around we?ll rotate our triangle again and make another one here and then we can complete our square and just for fun we?ll just complete the larger square around it. And so students can find the are of this entire square and subtract the triangles once again and end up with an area of twenty-nine square units. And so students can keep going and going and see how many different squares they can find and calculating their areas. And what they?re gonna be seeing is they?re gonna be doing the same procedure all over and over again. And that?s when we can start going into something more formal if students don?t discover it for themselves.

We?re going to build a model of what we were previously doing on the dot paper. And rather than starting with the square, we?re going to start with the triangle that we built our square on. We can do that many different ways and we could use a protractor to measure the ninety degree angle to build that. We could use a compass as well. And we could do the construction that way. Or we can even make a right triangle by using the one that is already on a piece of paper, which is what I?m gonna do because I need all the help I can get with arts and crafts.

And so here?s our perfect ninety degree angle and there?s our triangle.

Now what we?re going to do is we?re going to build the model that we saw on the dot paper. So we?re going to make four copies of our triangle and trace them on a new piece of paper. And by doing so we?ve created our square. Then we?re gonna color in each of the four triangles. This is because we are gonna be cutting these out later and we wanna distinguish those triangles from the square with a different color.

Now that we have that we?re gonna label this. Previously, we used numbers for each sides. But now we?re gonna use letters to represent any amount. So, we?re gonna call the long side of our right triangle A, were gonna call the short side B and our hypotenuse we?re going to call C. So, before we were choosing what A and B were and we never chose C. C is just kinda there and we don?t even know what C?s length actually is but that really doesn?t matter. Now as I label these we?re gonna label all four triangles on the inside of the triangle because we?re gonna cut these out and rearrange these and for that reason we are going to do all four triangles with R, A, B, and C. Now that we?ve done that the square that we wanted to calculate the are of in the center is built upon the side length of C. So now we can actually call that square C-squared and that?ll help us distinguish it from the other squares that we may have like the large square of the whole picture for example.

Now that we?ve done that we?re going to cut out that large square and were going to make a copy of it and we are going to do that by tracing that on another sheet of white paper. Once we have that done we?re going to cut out the four triangles and build the second model that looks exactly like the first one. So, now that we have our copy here we see that this C-square here is the exact same area as this square is over here. And what?s interesting about this square here, and the reason why we took these pieces and we copied them here is that if I take this triangle for example here and put it something crazy like this, I now have this really strange looking white area here. But since these triangles are still inside the square this white area is still the same as C-squared even though I don?t know how to calculate it directly, logically the area hasn?t changed. It?s still the same size. So what that tells us is that I can arrange anyway that we would like to. And here is how I would like to do this. So, I?m gonna take this piece and slide it up here. I?m gonna take this one and slide it here and then slide this one over here. And now this area that was this one giant square that was equal to C-squared is now arranged into two smaller areas and these happen to be squares as well. How do we know that they?re squares? Well the length of this square is A and the length of this side of this square is A therefore this is a square that is equal to A-squared. So whatever number we chose for this side, this is the square of that number. And down here we also have a square that?s length B by B so that makes this equal to B-squared. So we took that square that was in this form and we just rearranged it to look like these two squares here. So that tells us that this square plus this square together is equal in area to this square just by this rearrangement method. And what?s very cool is that we can see that this is our original triangle right here is that we wanted to find if we ignore everything else we just wanted to find the are of this square that was built on this diagonal here. And we can do that instead of doing anything else by subtracting a bunch of triangles we can simply find this square here and this square here and add their areas together to find this area over here. Which is what we wanted.

Something else that?s pretty awesome and the reason why I arranged it this particular way is that this should look familiar to students if they?ve come from Montessori because this is just a model of the binomial square. Where we have this length A plus B and this is A plus B squared with the A-squared, the B-squared, and the two rectangles obey B.

So as students start seeing that they?re doing the same thing over and over again, they?re finding the area of the large square and their subtracting away four triangles they might start to come to something like this. So for example in this one where they had the right triangle lengths of five and two, they might say okay the square, the big square is going to be length five plus two, and we always square that. So would give us forty-nine for the whole square. Then we are going to subtract away four triangles and each triangle?s area is going the be one-half the base which is two or five depending on how you look at it, five times two. So we have seven squared which is our large square minus two rectangles of ten and we get those if we put those triangles together to make two rectangles so altogether we have an area of forty-nine minus the twenty from the rectangles which gives us an are of twenty-nine square units.

But if we see students doing that type of work, we can say alright whatever length you choose we?re gonna call one side A and the other side B. and so the length of your square is going to be A plus B. then you?re gonna subtract four triangles. And the areas of those are one-half times A times B. So in reality what we have is A plus B squared minus two AB?s. Now this A plus B squared students may have seen before and if they come from Montessori elementary they would have seen this. And this represents A plus B squared right here where we have four parts. We have an A-squared, because this length here is A this length here is B so this is A plus B, so this is A-squared and we have to of these rectangles here and this is A by B, so this is AB plus another AB is two AB plus this B-squared in the corner minus two AB. And now we have positive two AB and minus two AB so that just gives us A-squared plus B-squared. So if we look back on our numerical example we had side lengths of five and two for our right triangle well if we just square five that gives us twenty five and if we square the two that gives us four and there?s our twenty-nine. So that?s a much faster way of going through all this because we understand that this process is gonna happen every single time. And that?s how we develop the formula. So what students have done through this is they?ve discovered that the square that?s built on the hypotenuse of that triangle is equal to the sums of the squares that are built on the two legs. They?ve discovered the Pythagorean Theorem.

So that?s it. That?s the Pythagorean Theorem. Now, the second way where the students were doing that numerical exploration and you could see that they?re kind of developing a formula. If we observe that then we can definitely go down that path with them and maybe ask them some questions so that they can discover that formula for themselves with the A plus B squared. However, a lot of students might not quite be there and so showing them the geometric interpretation, the one that Pythagoras did, that might be very intriguing for those students. And I think no matter what all the students enjoy seeing that. You can?t see a proof too many times and seeing it in different ways just add depth and richness to the work.

So, I hope you enjoyed the video and keep exploring the Pythagorean Theorem.

Description:

Michael Waski of Great Work, Inc.'s The Math Institute gives a lesson on The Pythagorean Theorem. To learn more about what The Math Institute can do to enhance your classroom experience, please visit: https://www.greatworkinc.org/adolescent-math-institute-1

Tags :
Michael Waski, Great Work, Inc., The Math Institute, Montessori Math, The Pythagorean Theorem, Montessori Adolescent, Math, Adolescent Math

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