NL2 PYTHAGOREAN THEOREM
This eighth grade mathematics lesson is an introduction to the Pythagorean theorem. It is the first in a sequence of five lessons focused on Pythagorean theorem. The lesson is 49 minutes in duration. There are 25 students in the class.
| Time | Caption |
|---|---|
| 00:00:03 | A lot. |
| 00:00:08 | Yes? |
| 00:00:08 | I really don't understand it. |
| 00:00:10 | Teacher, aren't we allowed to (inaudible)? |
| 00:00:11 | Yes. |
| 00:00:16 | Yes, yes, yes, yes. |
| 00:00:17 | I said just leave it. |
| 00:00:22 | Okay, ladies and gentlemen. Ravi, your punishment lines? |
| 00:00:24 | Oh, yes. |
| 00:00:26 | As first thing, before I forget again. |
| 00:00:27 | (inaudible) that too? |
| 00:00:29 | I have already shown that one. He already had that neatly signed off. The rest of you also grab your things. |
| 00:00:37 | Where are you coming- oh, you already went. Okay. |
| 00:00:46 | Wasn't that twice then? |
| 00:00:47 | Once. |
| 00:00:48 | Twice we were supposed to, didn't we? Oh. |
| 00:00:57 | Yup! Very good. Very good. |
| 00:01:05 | Okay. Um, like I said yesterday, today we will begin with Pythagoras. Which one of you ever heard of Pyth- Pythagoras? |
| 00:01:18 | Uh... |
| 00:01:19 | Uncle Pete. |
| 00:01:20 | Pete, Pete, Pete. |
| 00:01:21 | Tell us, what do you know about it? |
| 00:01:23 | It's a gentleman. |
| 00:01:24 | A squared plus B squared, equals C squared. |
| 00:01:25 | You know the equation A squared plus B squared equals C squared. Who else can say more about it? |
| 00:01:29 | Uh, I think this guy was a Greek. |
| 00:01:32 | The, uh, this gentleman was Greek. |
| 00:01:33 | Yup. |
| 00:01:34 | Or at least during that time period, somewhere in that region. |
| 00:01:37 | Okay. Does anyone know when you would use the Pythagoras' theorem and what you would use it for? |
| 00:01:43 | Yes, in a diagram. |
| 00:01:44 | In a diagram. You obviously have a diagram. I am crazy about diagrams. |
| 00:01:48 | (inaudible) |
| 00:01:49 | Something about triangles, indeed. Well, let's take another look. |
| 00:01:55 | What well-known triangles do we know? We have different types of triangles. |
| 00:01:59 | Right-angled. |
| 00:02:00 | We have a right-angled triangle. |
| 00:02:01 | Isosceles. |
| 00:02:07 | What is a characteristic of a right-angled triangle? |
| 00:02:10 | It has, it has one- |
| 00:02:13 | Ruben? |
| 00:02:14 | It has a right angle. |
| 00:02:15 | In any case, it has a right angle. |
| 00:02:20 | You can see that by the little symbol. What other kinds of triangles do we have? |
| 00:02:24 | An isosceles triangle. |
| 00:02:25 | An isosceles triangle. |
| 00:02:31 | What is characteristic of an isosceles triangle? |
| 00:02:34 | Two equal sides. |
| 00:02:35 | Two equal sides. Like this, for instance. I will just name the sides that are equal, for example A, the other one B. Meaning: |
| 00:02:47 | if I fill in a number for A, then I have to use the same number for every A I come across. |
| 00:02:52 | What else do I know about an isosceles triangle? |
| 00:02:57 | If I look at angles, for example? Maarten. |
| 00:02:58 | That A and B are equal. |
| 00:03:00 | A and B are equal in size. Very good. And I have another special triangle. Which one? |
| 00:03:07 | Equally shaped. |
| 00:03:08 | An equilateral triangle. |
| 00:03:09 | Oh. |
| 00:03:15 | What is characteristic of an equilateral triangle? Julie. |
| 00:03:18 | All sides are equal. |
| 00:03:19 | All the sides are equal, all three sides. |
| 00:03:29 | So one could say: all the sides are A. Would there be anything else I can mention about an equilateral triangle? |
| 00:03:35 | Um, all angles are equal. |
| 00:03:38 | All angles are equal. And can we tell how wide they are? |
| 00:03:41 | Yes, 180 divided by three. |
| 00:03:42 | One hundred eighty divided by three and that is? |
| 00:03:45 | Sixty. |
| 00:03:46 | Sixty. |
| 00:03:51 | Okay. So as we noticed before, we use Pythagoras in a triangle. But I have three types of triangles. |
| 00:03:59 | And we only use Pythagoras in a right-angled triangle. So we are going to continue with a right-angled triangle. |
| 00:04:06 | And the other triangles you will come across at some point later in the chapter. |
| 00:04:09 | Because you will be making right-angled triangles from these- or in these. Which enables you to cal- calculate things. |
| 00:04:19 | Another thing we noticed yesterday- |
| 00:04:23 | No, let me first pick up something else. I'll go- I'll return to the right-angled triangle. |
| 00:04:30 | Except I placed it on its side. |
| 00:04:33 | We know that the edges of a triangle- or any figure- are called "sides". |
| 00:04:38 | In a right-angled triangle, this side is attached to a right angle. So what should we call this side? A right-angled side. |
| 00:04:47 | Yes? Because this side is attached to a right angle so you call that a right-angled side. |
| 00:05:00 | Do we have any other right-angled side in there? |
| 00:05:02 | Yes. |
| 00:05:03 | Yes, all the way on the other side. That one is attached to the right angle as well, therefore you call that a right-angled side as well. |
| 00:05:19 | Then I still have one side left. It isn't so obvious because it is laying flat. But if you see this triangle, what can we call that side? |
| 00:05:28 | The long side. |
| 00:05:29 | The long side. That is correct. Or in a different way? |
| 00:05:33 | The right side? |
| 00:05:34 | It is actually at an angle. If you see it in such an- like a diagonal- so you call this the sloped side or the hypotenuse, is what you call this one. |
| 00:05:45 | These are just names, you know, you may also keep calling this "the long side", no problem. |
| 00:05:50 | Okay, with Pythagoras, soon, once we have gotten to know Pythagoras, we will use Pythagoras to calculate the sides. |
| 00:05:58 | Because all we know now about a triangle and the size of its side, is when we measure it. |
| 00:06:03 | However, those triangular protractors, how exact are they really? |
| 00:06:07 | We know from, uh, the drawing of your number line, from, for example, of the square root of five- |
| 00:06:14 | Well, you can't really measure that exactly. |
| 00:06:16 | Because the square root of five had such an odd number behind the point. And still we can calculate things in the end with Pythagoras. |
| 00:06:26 | Yesterday we saw, when I have a square, with a surface of, say, 37, |
| 00:06:35 | we could calculate, or at least determine how large the side was of such a square. |
| 00:06:40 | And we were actually able to do that quite precisely. Namely, Maarten? |
| 00:06:44 | The square root of 37. |
| 00:06:45 | The square root of 37. Okay, so that's "when I know the surface area of a square, how do I determine the sides again?" Okay, very well. |
| 00:06:59 | That is already kind of what's explained in assignment one. |
| 00:07:03 | This has to do with assignment two. That diagram is shown at assignment two. |
| 00:07:12 | We have here- I am just going to skip several parts of exercise two because we will not do all that. |
| 00:07:18 | We have a large square here: O E F G. As well as a square that is tilted at a corner: square A B C D. |
| 00:07:29 | I would like to know what the size the lengths are of the square A B C D, |
| 00:07:34 | for example what's the length of A B. We can't measure it and yet we are able to calculate it right now. |
| 00:07:41 | Because we have seen that once I knew the surface area of a square, I was able to determine the sides. |
| 00:07:46 | So I would like to calculate what the surface area is of the square A B C D. How can we do that? By which method? Rene? |
| 00:07:55 | Well, I think you have to up on top of the large square- oh, no, that's something you don't know- |
| 00:08:01 | Yes, though. |
| 00:08:02 | Yes, then you have to, uh, remove those angles- or, uh, yes, well, that's how you get the small square. |
| 00:08:07 | Very good. You have to calculate the surface area of the large square: O E F G. |
| 00:08:12 | And then you take the surface areas of the triangles, you subtract that. Very well. |
| 00:08:17 | What is the surface area of the large square, of the O E F G? |
| 00:08:28 | How much? |
| 00:08:30 | Uh, nine centimeters squared. |
| 00:08:31 | Nine- well, what the unit is I don't know, I didn't specify that. But if it were shown in centimeters, then you're right. |
| 00:08:37 | Right now I don't have a unit. Therefore the total surface area is indeed nine. |
| 00:08:40 | Namely three by three, or you can just count the squares- that gives us nine. |
| 00:08:45 | So we're told that we have to remove the surface areas of the triangles. What is the surface area of one of those triangles? |
| 00:08:50 | How do we calculate that again? |
| 00:08:54 | Willem? |
| 00:08:55 | I don't know how you calculate it but I can just see it. |
| 00:08:57 | You can just see it. What is the outcome according to you if you are just able to see it? |
| 00:09:01 | Um, four in total comes off because one of those, uh... |
| 00:09:06 | One of these triangles? |
| 00:09:07 | That is one. |
| 00:09:08 | That is one. Okay. |
| 00:09:09 | And there are four of those. |
| 00:09:10 | Correct, but how are you able to calculate that again? Leni? |
| 00:09:14 | You make it into a rectangle? |
| 00:09:16 | I- I am just going to pull it aside. So these are then the three surface areas of the triangle that we want to calculate. |
| 00:09:24 | We will make it into a rectangle. And then? |
| 00:09:29 | We calculate the surface area. |
| 00:09:31 | We calculate the surface area. That was one by two, so the whole surface area together of the rectangle is then two. |
| 00:09:37 | And I just need to take half of that, which brings us to one, very good. |
| 00:09:46 | So the surface areas of the triangles- |
| 00:09:54 | So the surface areas of the triangles together here is four. So what is left is five. |
| 00:10:05 | So the surface of the square in the center, A B C D, is five. |
| 00:10:12 | And now we can finally, after all that calculating, determine one of those sides. Namely, that is? |
| 00:10:18 | The square root of five. |
| 00:10:19 | The square root of five. Very good. |
| 00:10:24 | So we've got no problem figuring this out. We may not even need Pythagoras at all. |
| 00:10:28 | No. |
| 00:10:29 | But it is kind of a hassle if you constantly have to draw on like this. |
| 00:10:32 | Because, suppose you wanted to know the length of this little line. |
| 00:10:38 | If you want to calculate that you first have to, you first have to draw a square again, like this one and another square around it. |
| 00:10:44 | And then calculating it again. So it's a real hassle. So we're going to do that differently. |
| 00:10:50 | The example given in the book is that assignment four and five. I will just do that one with the overhead projector. |
| 00:11:00 | :00] |
| 00:11:14 | Okay, we see, we see at assignment four, they have given two plates. Well, I drew those too, I have two plates. |
| 00:11:24 | They should be equal in size. |
| 00:11:26 | Well, if I place them on top of one another, they are indeed- if I place them very neatly on top of each other, equal in size. |
| 00:11:33 | Yes. Yes. |
| 00:11:35 | Well... |
| 00:11:36 | Approximately. |
| 00:11:37 | They are really equal in size but they stick a bit. |
| 00:11:40 | Yes. |
| 00:11:41 | There. Yes? Convinced? |
| 00:11:43 | Yes. |
| 00:11:44 | Very good. Yes, of course there are some who will never be convinced. |
| 00:11:48 | I said yes. |
| 00:11:49 | Yes, but I also heard a no. |
| 00:11:56 | Then we also see- here they made eight right-angled triangles. |
| 00:11:59 | Namely, four right-angled triangles in the one board and four right-angled triangles in the other board. |
| 00:12:05 | I cut those as well. Those are also equal in size. When I stack them on top of each other they're all equal in size. |
| 00:12:10 | So it's truly a stack of eight and I will take them off from there. |
| 00:12:14 | We will lay it down just like it is shown in the diagram in the book. |
| 00:12:17 | Which question? |
| 00:12:18 | Question four is what we are working on. |
| 00:12:24 | :00] |
| 00:12:32 | Very well, we are now able to, in the same way as exercise- what? Is one lying incorrectly? |
| 00:12:37 | No. |
| 00:12:38 | Oh? Yes, it's crooked- I mean it is loose. |
| 00:12:42 | (inaudible) |
| 00:12:45 | Now, should we know the size of everything, we could calculate what the area is of the square in the middle here. |
| 00:12:51 | Namely, in the same way that we did over there. |
| 00:12:54 | The surface of the whole board and then the surface area of those three- of those four triangles taken off. We won't do that. |
| 00:13:01 | Now I will, with the other four triangles I have, I will make the other one. |
| 00:13:11 | It lies in this corner over here, and it lies in that corner over there. |
| 00:13:22 | Those triangles- those four triangles that are lying over here, are equal in size as the triangles that are lying over there. |
| 00:13:27 | Yes. |
| 00:13:28 | I knew that the board is equal in size. What am I able to say about the surface of this square and that square? |
| 00:13:34 | They are equally big as the other ones. |
| 00:13:36 | Together they are- if I add these two surface areas together- are equal in size as that one. Yes? |
| 00:13:41 | Because the boards are equal in size. And the triangles are equal in size. Okay... |
| 00:13:47 | What they indicate in the book, then, is that they are going to put those boards on top of each other. Yes? |
| 00:13:52 | Well, you can't see through the board but you can through the overhead sheets. |
| 00:13:55 | So I will just remove that then because in the book it is shown underneath it. And I will try to place it on top of it. |
| 00:14:03 | Approximately. I may not move anything. There. |
| 00:14:08 | We are still aware that the surface area of this square is equal in size as the surface of this square plus that square together. |
| 00:14:15 | Because they are still the same boards and they are still the same triangle. |
| 00:14:21 | What are we going to say now? What are we going to look at? Imagine that this side- that that is 12, |
| 00:14:23 | then what is the surface area of the square that is attached to it? |
| 00:14:30 | Twelve times 12. |
| 00:14:31 | Twelve times 12. And that is? |
| 00:14:32 | One hundred forty-four. |
| 00:14:33 | One hundred forty-four. |
| 00:14:34 | Without a calculator. |
| 00:14:35 | Oh yes. |
| 00:14:37 | The short side, that is five. |
| 00:14:40 | Twenty-five. |
| 00:14:41 | So it becomes 25 indeed. What do I know about the large square, then? |
| 00:14:47 | One hundred forty-four plus 25. |
| 00:14:48 | That 144 plus 25 and that is? |
| 00:14:53 | (inaudible) |
| 00:14:54 | One hundred sixty-nine. |
| 00:14:57 | If I know that the surface area of the large square is 169, can I say something about how long this sloped side is? |
| 00:15:04 | Which angle? |
| 00:15:05 | Of this- of this side of the square. It is also the side of the triangle. Namely? |
| 00:15:10 | Half of 12. |
| 00:15:11 | Twelve. |
| 00:15:12 | The square root of 169. |
| 00:15:13 | The square root of 169. Why should you take half of 12? |
| 00:15:20 | On this side, I have this triangle lying over here. The one side is 12, the other side is five. I want to know the hypotenuse. |
| 00:15:27 | I knew that the surface areas of the squares that are lying here- that have exactly the same sides as the sides of the triangle- |
| 00:15:35 | if it'd be so kind as to stay on it's spot- |
| 00:15:39 | is equal in size as the surface area of the large square. Yes? |
| 00:15:42 | We have seen that the surface area of the large square is 169. |
| 00:15:46 | Now I would like to know one side, that is the side of the square, which is at the same time also the side of this, of this triangle, |
| 00:15:54 | which is indeed the square root of 169, which is 13. |
| 00:16:01 | Yes? And that is actually sort of- of what Pythagoras was talking about. |
| 00:16:06 | Pythagoras says that if I place a square on the one right-angled side, |
| 00:16:11 | so the surface area of the square I will calculate. I will place another square on the other- the shorter side. |
| 00:16:19 | I will take the surface area of that too, of that square. If I add these two surface areas together, |
| 00:16:24 | apparently- he discovered all this- they are exactly the same as the surface area... |
| 00:16:32 | of the square that is placed on the hypotenuse. Yes? |
| 00:16:36 | I'll show that one to you in a second. As I will write the general rule on the board. |
| 00:17:26 | We have seen- ladies and gentlemen, we are going to continue. Ravi. Rick! |
| 00:17:36 | Okay. This is the same as what was just on the screen. Suppose that this is the triangle of which this side is 12. |
| 00:17:46 | Then we would know that the surface area of this square was 144. |
| 00:17:50 | If this surface area is five, then this one is 25. |
| 00:17:55 | This one was 144 plus 25, is then 169. |
| 00:18:00 | So the length is 13. |
| 00:18:02 | So this one is 13. That was also what we just had on the screen. But now we are going to construct very generally- |
| 00:18:06 | Now I get it! |
| 00:18:10 | It is also actually in the book, except I am writing it down a bit more detailed. |
| 00:18:13 | So perhaps it would be smart if you copied it. Then you can always retrieve it. |
| 00:18:34 | Yak! |
| 00:18:37 | Ladies! |
| 00:18:43 | The Pythagorean theorem. What have we done? We had a right-angled triangle. It only applies to right-angled triangles. |
| 00:18:52 | One of those right-angled sides, let's take 12 as the first right-angled side. What did we do with this right-angled side? |
| 00:18:57 | We took the square of this right-angled side. In fact we glued a square over it and calculated its surface. |
| 00:19:04 | So you took one right-angled side, squared. So we will write that down: one right-angled side, squared. |
| 00:19:11 | Are you supposed to- |
| 00:19:13 | Yup. |
| 00:19:14 | You are supposed to copy that thing, right? |
| 00:19:16 | What I am writing down you have to copy. |
| 00:19:18 | Oh. Ah. |
| 00:19:26 | So we had: one right-angled side squared, that is, in other words, the surface area of the square. What did we do with that? |
| 00:19:37 | We added the surface area of the square that was on the other right-angled side to it. |
| 00:19:41 | How did we determine the surface area of the other square again? |
| 00:19:44 | So that is plus- taking the other right-angled side squared. |
| 00:20:08 | So we had the one right-angled side squared, which is in fact the surface area of the square that you attached to it. |
| 00:20:13 | Plus the other right-angled side squared: the surface area of the other square that you attached to it. |
| 00:20:19 | What is it equal to? To the surface area of the large square that is on the hypotenuse. |
| 00:20:24 | So actually, surface area is the same as that side squared. |
| 00:20:31 | This is: the hypotenuse. |
| 00:20:39 | :00] |
| 00:20:50 | So that is the hypotenuse squared. Does that thing bother you? Sorry. Since I won't be needing this one anymore. |
| 00:21:05 | There. |
| 00:21:10 | The table is slanted. |
| 00:21:14 | So this is the general theorem. Let's see if it will work if we apply it. |
| 00:21:32 | Gentlemen! Come on. |
| 00:21:39 | By now you guys must know that I am crazy about diagrams and tables and things like that. |
| 00:21:44 | Yes, we knew that because you already told us yesterday. |
| 00:21:46 | There we go. There we go. Good thing the book utilizes that too. Otherwise I would have to explain it all by myself once again. |
| 00:21:52 | But the book utilizes it too. So we are going to- to calculate those sides, we will also use the diagram. |
| 00:21:59 | Well, if you copy this, Ravi, it'll speed up your homework for you will have done this assignment already. |
| 00:22:03 | (inaudible) fast hey (inaudible). |
| 00:22:24 | Okay. Here I have a right-angled triangle of which I already know two of the right-angled sides. The hypotenuse I don't know yet. |
| 00:22:32 | That's what I want to calculate. We will make a small table. I named it "side" on one side. |
| 00:22:39 | As in: how long is the side going to be that I am about to fill in- Rick? |
| 00:22:41 | You know you can also proceed in the hallway if you're so sure you understand it. |
| 00:22:48 | And the squared value actually represents the surface area of the square that you placed on the sides. |
| 00:22:55 | Yes. |
| 00:22:56 | What? |
| 00:22:57 | So here I have- what were we supposed to do? The one right-angled side. "The right-angled side" I will abbreviate. |
| 00:23:03 | Because otherwise you will keep writing. I need the one right-angled side; I need the other right-angled side. |
| 00:23:15 | Yes? |
| 00:23:16 | Ladies, would you please just be a little more quiet as well? |
| 00:23:19 | They are filming in this chicken coop. |
| 00:23:22 | That doesn't matter. I just want it quiet for myself. |
| 00:23:26 | And we have a hypotenuse. Yes? |
| 00:23:31 | So what did we see? We were going to take the one side and square it because that was in fact the one with the square attached to it. |
| 00:23:36 | And you were going to determine the surface area of that. That one side is in this case, for example, three. |
| 00:23:41 | What is- is that squared? Or in other words, what is the surface area of the square that you attach to it, Julie? |
| 00:23:44 | Nine. |
| 00:23:45 | That is nine. |
| 00:23:46 | Julie? |
| 00:23:47 | Julie? Am I- |
| 00:23:49 | She is sick! |
| 00:23:50 | Oh, she is sick, yes, well then people shouldn't move around like that. Sorry, Myrte. |
| 00:23:54 | The other right-angled side is four, so that is squared. |
| 00:24:01 | Now you should listen very carefully because this is a mistake that is often made. |
| 00:24:05 | We had said, the one right-angled side squared plus the other right-angled side squared. Those are the only ones you may add together. |
| 00:24:10 | So only the squared sides. So not your regular sides. |
| 00:24:16 | So you are allowed to only add this side. And that gives you? |
| 00:24:22 | Twenty-five. |
| 00:24:24 | Oh, that is four times four and three times three and then subtract- |
| 00:24:26 | Then the other one is five. |
| 00:24:28 | And then I would indeed want to know: how long is the hypotenuse. |
| 00:24:32 | You know that the 25 represents the surface area of this square up here. |
| 00:24:36 | Or, in other words, you are going to take the square root, this way. And that leaves me with five. |
| 00:24:43 | Yeah. I already knew that. |
| 00:24:45 | Any questions about this? |
| 00:24:47 | Yes, but it is- no. |
| 00:24:49 | No, no questions? Regardless that you say: "yes, but". |
| 00:24:54 | Oh, like that, yes, yes I got it. |
| 00:24:56 | Yes, do you? Myrte? |
| 00:24:58 | But it says, uh, "the other right-angled side equals the hypotenuse squared", but do we have to take the square root of it instead of, uh, squaring it? |
| 00:25:07 | Yes. The general equation therefore is- uh, Rick, could you please go into the hallway now? |
| 00:25:13 | Bye! |
| 00:25:23 | What it says here is "one right-angled side squared plus the other right-angled side squared equals the hypotenuse squared". |
| 00:25:31 | Therefore, if you want to know what the hypotenuse is, then you have to take the root of the number representing the hypotenuse squared. |
| 00:25:38 | Is it squared and then you have to- yes, okay. |
| 00:25:40 | This whole table over here shows these numbers squared. |
| 00:25:42 | Yes. |
| 00:25:43 | Yes? Very well. |
| 00:25:44 | You figured it out. |
| 00:25:45 | Well, in this way you can also reconstruct... |
| 00:25:52 | In this way you can also reconstruct how the A squared plus B squared equals C squared is found- gentlemen, |
| 00:26:00 | I wasn't quite finished yet. You can start shortly. |
| 00:26:06 | If I have a right-angled side of which I call the one side A, the other side I call B and the hypotenuse I call C, |
| 00:26:13 | and you apply the equation, then indeed you will get my first right-angled side squared. |
| 00:26:19 | For example my first right-angled side is A, and you take that one squared. Plus my other right-angled side, take that one squared as well. |
| 00:26:28 | Equals your hypotenuse squared. And that is how most of your parents learned this. |
| 00:26:33 | Oh! |
| 00:26:34 | They never had it explained to them with squares. They just learned it kind of like "this is how it is so you better get to work with it". |
| 00:26:42 | A squared plus B squared equals- |
| 00:26:46 | Oh, right. |
| 00:26:49 | Okay. |
| 00:26:50 | (inaudible) shows the thing squared there? |
| 00:26:52 | Because that notion squared is the surface area that you want to calculate. The surface area of the square. |
| 00:27:01 | Like that would help! |
| 00:27:08 | Okay, ladies and gentlemen. As far as the homework is concerned. Suzanne? |
| 00:27:14 | Jeffrey, Maarten? |
| 00:27:19 | So far, by way of what I showed you on the board- and I am hoping that you guys did copy it- |
| 00:27:26 | we have already completed quite a lot of the exercises. |
| 00:27:28 | We did exercise one and we did exercise two. We haven't done exercise three yet. That's homework... |
| 00:27:41 | Hey! You are being filmed! |
| 00:27:53 | They will just edit that out. |
| 00:27:54 | That's right. Each time Ravi's head will be cut out. Yes. |
| 00:28:07 | You guys still have 20 minutes left. |
| 00:28:08 | Twenty. Yes. Yes. |
| 00:28:10 | That means that you have to- of these exercises, be able to get as far as exercise 11. Presumably even further. |
| 00:28:17 | These we haven't all done yet, those are new. The only advice I'll give you on exercise 11, a lot is written down about triangles there- |
| 00:28:25 | Sorry, Willem- |
| 00:28:26 | The only thing (inaudible). |
| 00:28:27 | Yes, it is fine like that. |
| 00:28:28 | Yes? So with exercise 11 you will only get data on what angles are 90 degrees and what the lengths are of certain sides. |
| 00:28:37 | Half of you are now missing this instruction. |
| 00:28:38 | Oh, oh, sorry, you have it right here. |
| 00:28:40 | Annelies and Myselle? You girls are missing the instruction. |
| 00:28:44 | Shh. |
| 00:28:50 | Exercise 11 only states what angle is 90 degrees and what the sizes are of particular sides. |
| 00:28:57 | If you want to know exactly how things are put together, make a diagram, which explains the comment at the bottom; |
| 00:29:02 | make diagrams of those triangles at exercise 11 so that you know what data you've got. |
| 00:29:07 | Either both of the right-angled sides will be given, or perhaps only the right-angled side and the hypotenuse will be given. |
| 00:29:15 | So that you can complete your calculation. Yes? |
| 00:29:18 | Yes, understood. |
| 00:29:20 | For those of you who want to check, I have answer booklets over here. You can grab those and get started now. |
| 00:29:38 | From now on I really want it to be quiet when I am instructing. |
| 00:29:39 | Yes. |
| 00:29:40 | Yes? |
| 00:29:41 | Yes. |
| 00:29:42 | You are one of the few- or one of the few, together with a few more- but there are several who want nothing but to pay attention. |
| 00:29:46 | Yes. |
| 00:29:47 | If you keep on disrupting, then you are disrupting everyone. |
| 00:29:48 | Yes. |
| 00:29:49 | And certainly me. |
| 00:29:50 | Yes. |
| 00:29:51 | Just quietly get to work, the homework is on the blackboard. |
| 00:29:56 | :00] |
| 00:30:12 | Just put it on, uh... |
| 00:30:13 | It doesn't make any sense, you don't feel (inaudible). |
| 00:30:18 | Yes, no, don't close it completely, just put it on- I do want some oxygen to come in. |
| 00:30:21 | Oxygen? |
| 00:30:22 | The heater is still on. |
| 00:30:23 | Well, I can't turn it off, it is central heating. Otherwise I would be the first one to turn it off. |
| 00:30:27 | Here: two plus four plus six. |
| 00:30:39 | Uh, ladies, ladies and gentlemen, when you are working I would like it to be more quiet. Discussions are fine but please do it in a whispering voice. |
| 00:30:52 | :00] |
| 00:30:57 | He has the best-looking notebook in the whole class, of course. |
| 00:31:00 | Beautiful notebook. |
| 00:31:11 | How can you determine how much of it is tiled? |
| 00:31:14 | What does it look like, the example, the diagram? How does the whole diagram look, the whole garden? |
| 00:31:20 | As a square. |
| 00:31:21 | It's an exact square. |
| 00:31:23 | Yes? |
| 00:31:25 | Yes? And what is the shape of that tiled part? |
| 00:31:28 | A triangle. |
| 00:31:29 | A triangle. How could we calculate again the surface area of such a triangle? |
| 00:31:35 | We just did that on the board on the other side. |
| 00:31:37 | Yes, I already have it! |
| 00:31:39 | Ladies and gentlemen, the fact that I have my back turned towards you doesn't mean you are allowed to turn things upside down. |
| 00:31:45 | I agree. |
| 00:31:51 | How did we do that? What can you make of this, of that one triangle? |
| 00:31:54 | A, uh, rectangle. |
| 00:31:56 | A rectangle. Yes? You can make a rectangle of this. What is the surface area then of that rectangle? |
| 00:32:04 | Eight times six is 48. |
| 00:32:05 | Forty-eight. And what part of the triangle is it? |
| 00:32:09 | I don't know. |
| 00:32:11 | Considering you know that the whole rectangle- because you said it's six times eight- |
| 00:32:13 | This one is 24. Twenty-four squares (inaudible). |
| 00:32:15 | Correct, is half of it. Yes? |
| 00:32:18 | Yes. |
| 00:32:19 | Okay. |
| 00:32:22 | Shh! Whisper, Maarten. Maarten, whisper! |
| 00:32:25 | I don't know it anymore. |
| 00:32:27 | Ladies should be whispering as well. |
| 00:32:29 | I still don't understand G eight. |
| 00:32:32 | No, right. Okay, oh. Come over here for a second. |
| 00:32:39 | :00] |
| 00:32:47 | G eight. |
| 00:32:50 | What does G eight say? We have a tiler, who uses white as well as blue tiles. |
| 00:32:55 | Yes. |
| 00:32:56 | Yes? The surface area of every tile is one square decimeter. The white ones as well as the blue ones. |
| 00:33:03 | Yes. |
| 00:33:04 | Okay. Five tiles together make the square A B C D. A, B, C, D. What is the surface area of that square? |
| 00:33:14 | Um, (inaudible). |
| 00:33:17 | Sorry? |
| 00:33:18 | Twenty-five. |
| 00:33:19 | Why 25? We are only looking at this one. |
| 00:33:30 | You know that every tile, the white one as well as the blue one, is one square decimeter. |
| 00:33:33 | Yes. |
| 00:33:34 | Yes? |
| 00:33:35 | Five. |
| 00:33:36 | So it's five, because there are five tiles in there. Yes? The surface area of this is five. |
| 00:33:41 | What is the length then of A B, given the surface area of that square is five? |
| 00:33:49 | The square root of five. |
| 00:33:50 | The square root of five. Yes? Maybe later you should add for yourself an ex- a written explanation on how you got that square root of five, okay? |
| 00:33:57 | Because if later on you- you read it again you won't know how you get that square root of five. |
| 00:34:00 | Yes. |
| 00:34:01 | Yes? Now we have to fill in the length of A E, A E goes from here to there. |
| 00:34:07 | Is two times AB . |
| 00:34:08 | Yes, is two times A B. Because that section is just as long as this. And that is, then, twice? |
| 00:34:14 | (inaudible) |
| 00:34:15 | Yes, very good, two times the square root of five. |
| 00:34:20 | Very good. No, no, no, no sorry, no. We've seen that yesterday. How can we write the two differently? |
| 00:34:25 | But you don't have to do that here yet, that will come later on. |
| 00:34:28 | (inaudible) to write the two differently? |
| 00:34:31 | How can I write the two into a square root? What is two equal to again? |
| 00:34:36 | Yesterday we had that whole list on the board. |
| 00:34:39 | You knew that the square root of 16 was equal to four? |
| 00:34:42 | Yes. |
| 00:34:43 | Yes? |
| 00:34:44 | Oh, uh, it is equal to, uh, the square root of four. |
| 00:34:45 | Correct, equal to the square root of four, very good. So you have the square root of four times the square root of five here. |
| 00:34:52 | Look, the two was correct, but- |
| 00:34:53 | Oh, yes. |
| 00:34:54 | It says the same. What is the outcome of this then? |
| 00:34:58 | Yes! Yes, perfect. Very good. Try to do the rest of them now, too. |
| 00:35:03 | Yes. |
| 00:35:04 | Yes? |
| 00:35:06 | Look, I have a question, since yesterday I had something from chapter one- |
| 00:35:09 | Oh, yes. |
| 00:35:10 | And, um, look, this is, for example, um, the regular square- |
| 00:35:13 | Yes? |
| 00:35:14 | And that one is 40 square centimeters. |
| 00:35:18 | So the reduction goes to 10 square centimeters. Then the factor isn't four, is it? |
| 00:35:22 | No. |
| 00:35:23 | It is two, isn't it? |
| 00:35:24 | Correct, because the surface area becomes four times smaller- |
| 00:35:26 | Yes. |
| 00:35:27 | But your magnifying factor will be the square root of that. |
| 00:35:31 | So if you have, for example, a really large number, for instance, uh- well, yes, I don't know, uh, for example with five numbers- |
| 00:35:35 | Yes? |
| 00:35:36 | And, uh, it becomes a lot smaller, you have to divide it by four? |
| 00:35:40 | So, uh, the magnifying factor with reduction has to be divided by four, whereas if you make it from small to large it is times four? Okay. |
| 00:35:47 | Yes, correct. You were first. |
| 00:35:50 | I just have a question. |
| 00:35:51 | Yes? |
| 00:35:52 | Uh, could it also be that you only get this number? |
| 00:35:55 | Yup. Well, not only that, you need two sides at least, otherwise you cannot calculate anything. Yes? |
| 00:35:59 | So this one, and that one, and then all you have to do is calculate this one here? |
| 00:36:02 | Yes. Very good. |
| 00:36:03 | Okay. |
| 00:36:04 | Great. |
| 00:36:05 | Where is this key on my pocket calculator? |
| 00:36:07 | Oh, it is fine, right? |
| 00:36:09 | Here. Only yours doesn't have an X below. And this is your root. |
| 00:36:11 | So it's the same thing. |
| 00:36:12 | Yes. This is your root. |
| 00:36:13 | Okay. Yes. |
| 00:36:37 | Was it clear for you guys? |
| 00:36:38 | Yes. |
| 00:36:39 | Yes? Very good. |
| 00:36:44 | You keep track of this a little too, I hope? |
| 00:36:45 | Yes, yes, I just wrote it down. |
| 00:36:46 | No, okay. Oh, okay. |
| 00:36:47 | That's just the nature of the assignment. |
| 00:36:58 | Is this one clear to you guys? |
| 00:36:59 | Yeah, sure. |
| 00:37:00 | Really? |
| 00:37:01 | Yes with the explanation earlier it made it, in any case, clearer. |
| 00:37:02 | But in the beginning it seemed a bit weird, but now I understand it. |
| 00:37:03 | Yes, very good. So it is very important that it is written down properly here. Yes? |
| 00:37:09 | I mean you copied it here very nicely but try to retain some of that for yourself too. |
| 00:37:13 | Technically this doesn't get explained until here. So if afterwards you try to calculate the numbers, make sure you use it. |
| 00:37:19 | So don't try to be stubborn or anything like that. So really try to stick to the rules, those steps. Yes? |
| 00:37:24 | Yes. |
| 00:37:25 | Yes? |
| 00:37:26 | Yes, okay. |
| 00:37:29 | The same problem. |
| 00:37:32 | Three you can still do, in principle, in the old method. Because this one, the one I explained won't be explained until the next paragraph. |
| 00:37:40 | But it is allowed, you'll still work through it. |
| 00:37:41 | And, uh, the square root of (inaudible). |
| 00:37:45 | The square root of six. |
| 00:37:46 | Yes. |
| 00:37:47 | Yes, very good. |
| 00:37:48 | Yes? But if you want to calculate that one, what do you need then? |
| 00:37:52 | Then I can- just this one- like that? And, uh- |
| 00:37:55 | Yes, you can that way, very good. Yes, of course, only it hasn't been explained over here yet. |
| 00:38:00 | But since I have already explained it and as you understand it, then, as far as I am concerned, you can go ahead and use it. |
| 00:38:03 | Yes. |
| 00:38:08 | Yes? |
| 00:38:09 | If you want to measure this side- |
| 00:38:12 | Yes? |
| 00:38:13 | Then you have to- this is 225. |
| 00:38:15 | Yes. |
| 00:38:16 | And this is 64. |
| 00:38:17 | Yes. |
| 00:38:18 | Then aren't you supposed to take two- 225 plus 64- |
| 00:38:21 | Yes. |
| 00:38:22 | Is 289. |
| 00:38:23 | Yes, indeed, that square- |
| 00:38:24 | That you have to make into a square root? |
| 00:38:26 | Yes, very good. |
| 00:38:27 | So that becomes 17, then. |
| 00:38:28 | And that is 17. |
| 00:38:29 | Yes, perfect. Are you writing down what you are doing, though? Not only, uh, writing down the answers? |
| 00:38:33 | Oh. |
| 00:38:39 | Ehm, look here. |
| 00:38:44 | Can you see it? |
| 00:38:45 | Yes, I can see it. |
| 00:38:46 | One hundred ninety-six square meters. |
| 00:38:48 | What is this? What exercise are you working on? |
| 00:38:51 | Oh, with, um, three. |
| 00:38:52 | Yes? And what is this 196? |
| 00:38:55 | That is the whole garden. |
| 00:38:56 | Okay, the whole garden. Yes? |
| 00:38:58 | Then I have to do minus- |
| 00:38:59 | Yes. |
| 00:39:00 | Twenty-four square meters. |
| 00:39:01 | Yes. |
| 00:39:02 | But doesn't that just become- can you just take this off then, like, like in the same way if there wasn't a square meter on here? |
| 00:39:10 | Yes, because- the whole garden is in square meters- |
| 00:39:13 | I must be confusing it with something else then. |
| 00:39:14 | Very small sections of square meters. |
| 00:39:15 | So you can easily subtract square meters from square meters. |
| 00:39:17 | Now I don't understand any of this anymore. |
| 00:39:19 | Here I have the table, yes? There are square, well- let's just say meters. If I chop a piece off here, then this piece is also in square meters. |
| 00:39:29 | Then this piece will remain, in square meters. That is the same thing you will do with the garden. |
| 00:39:34 | Yes, fine. |
| 00:39:35 | Yes? |
| 00:39:42 | Everything is clear for you guys? |
| 00:39:43 | (inaudible) |
| 00:39:45 | Okay. |
| 00:39:46 | This is how it should be done? Like this? |
| 00:39:49 | Yes, perfect. Perfect, very nice. |
| 00:39:51 | Of course not, are you crazy or something? |
| 00:39:54 | But you were too warm weren't you? |
| 00:39:56 | Yes, right, I will just go and sit here without my T-shirt, in my bare chest. |
| 00:40:03 | Well, if the wind is gone, if the rain is gone then it can be opened again. I mean, uh... |
| 00:40:08 | Then it'll get even colder. |
| 00:40:10 | Are you cold? |
| 00:40:11 | Yes. Very badly! |
| 00:40:12 | (inaudible) only wearing this really thin thing. |
| 00:40:21 | Yes. You aren't going to infect your sister, are you? |
| 00:40:24 | No, she is, she is already sick. |
| 00:40:26 | Yes? She is at home, though? |
| 00:40:28 | Yes. |
| 00:40:32 | And have you changed her yet, or are you going to leave that to... |
| 00:40:37 | Because I did that by myself- |
| 00:40:39 | Yes? |
| 00:40:40 | I am allowed to carry her into the church for the baptism. |
| 00:40:41 | Oh, how neat. Oh, you don't like that? |
| 00:40:44 | Well, I do think it is kind of neat and all- |
| 00:40:45 | Yes! |
| 00:40:47 | Wow! |
| 00:40:52 | Well, wonderful... |
| 00:40:53 | Really, uh, great. No, I think it is kind of neat. |
| 00:40:56 | Yes. |
| 00:40:57 | Do you have the answer booklets? |
| 00:40:58 | Yes. |
| 00:41:06 | Shirley isn't here and uh... |
| 00:41:07 | Sixty. |
| 00:41:10 | A B. |
| 00:41:11 | No, she was sitting on Shirley's seat and that's confusing me again. I thought I finally knew all the names. Okay, thank you. |
| 00:41:21 | And who normally sits next to? |
| 00:41:24 | Sander. |
| 00:41:25 | Oh, yes. |
| 00:41:26 | Four times the square root of 20 is eh- |
| 00:41:28 | De Boer? Thank you. Easy? They were, weren't they? |
| 00:41:35 | One, two, three, four, five, six, seven... eight, nine, 10- 20. One, two, three, four, 25, 26, 27- |
| 00:41:41 | No? Yes though, it's 28, 29, it is correct. |
| 00:41:44 | Are you going to proceed with the work? |
| 00:41:49 | Can it be a little more quiet, uh, Jeffrey? And Maarten? |
| 00:41:53 | We just have to calculate this one, so we have to take half. |
| 00:41:58 | And then? |
| 00:42:00 | Well, then- |
| 00:42:01 | Will you indicate how you got the numbers? |
| 00:42:02 | Oh, yes, I will- |
| 00:42:03 | Yes? Because once again with the test I had a lot of- actually, I haven't seen yours yet, but from the other classes- |
| 00:42:08 | I had to mark a lot incorrect because they only had put answers down. |
| 00:42:14 | :00] |
| 00:42:34 | Very well. My handkerchief will go into the laundry tonight. I'll need it again in a minute. |
| 00:42:41 | Sixty-four. And then this you have to add together- |
| 00:42:45 | No. |
| 00:42:46 | Which makes 89. |
| 00:42:47 | No, they have been here on Tuesday and now. |
| 00:42:51 | What class was that then? |
| 00:42:53 | On Tuesday it was A 2 C. |
| 00:42:56 | With Dennis. |
| 00:42:57 | Yes. |
| 00:43:05 | Teacher? |
| 00:43:06 | Yes? |
| 00:43:07 | Can you check exercise three for a moment? |
| 00:43:09 | Of course. |
| 00:43:10 | Because if I have that one correct then- |
| 00:43:11 | Then you understand it, you mean. |
| 00:43:12 | Yes, because I am not sure if I understand it. |
| 00:43:13 | Yes. Except you have already used the new method which is explained to you in the next paragraph. |
| 00:43:18 | But if you understand it, then it doesn't matter at all. |
| 00:43:19 | No, I took the old method, I think. Oh, no! That is the new one. |
| 00:43:22 | Yes, that one won't be explained until the next paragraph. |
| 00:43:24 | Oh, so according to the book, I am actually working ahead. |
| 00:43:27 | Yes, but that doesn't matter because the answer should be the same. All right? |
| 00:43:30 | Here it says: how big is the (inaudible) and here I've got five, but do you just have to (inaudible)? |
| 00:43:37 | Yes, that is correct. Because the surface area of the square is five. |
| 00:43:39 | Yes. |
| 00:43:40 | So the side, that is just one side- |
| 00:43:42 | Yes. |
| 00:43:43 | Is the root of this. |
| 00:43:44 | Oh, yes. |
| 00:43:45 | Yes? That is what we saw several times yesterday. |
| 00:43:47 | (inaudible) |
| 00:43:49 | Sorry? |
| 00:43:50 | This square over here has four points- |
| 00:43:52 | You may draw it- |
| 00:43:53 | (inaudible) draw it? |
| 00:43:54 | No, you're not obliged to, no, you're not obliged. But if it clarifies things for yourself, then you may, of course. |
| 00:43:58 | But you don't have to. |
| 00:44:02 | I got it right. |
| 00:44:03 | Yes, very good. |
| 00:44:06 | You can check it this way. |
| 00:44:08 | (inaudible) |
| 00:44:13 | It should, I believe. But what, why? |
| 00:44:15 | How does one calculate this? |
| 00:44:17 | Surface area is square A E F G. How does A E F G go? |
| 00:44:22 | Okay, well, what is the surface area of that? |
| 00:44:26 | How much is, what, what is the surface area again of one of these angles? |
| 00:44:28 | Twenty. |
| 00:44:30 | No, the surface area. |
| 00:44:31 | Five. |
| 00:44:32 | Five. And how many of those little squares do I have in here? |
| 00:44:35 | Four. |
| 00:44:36 | Yes, so? |
| 00:44:37 | Twenty? |
| 00:44:38 | Twenty. |
| 00:44:39 | Yes, but I thought (inaudible). |
| 00:44:41 | Yes, that is correct. Because they are determining how you can write things in a different way. |
| 00:44:45 | Okay. |
| 00:44:46 | What I explained yesterday on how you can write out those roots in different ways. Yes? |
| 00:44:50 | Yes, okay. |
| 00:44:52 | Heinjan and I are going to BEVO shortly. |
| 00:44:54 | Yes, fine. |
| 00:44:55 | We have a test to take. |
| 00:44:56 | Yup. |
| 00:44:58 | Yes? |
| 00:45:03 | Hey, hey, Ravi you still have to, uh, take the test as well. |
| 00:45:05 | May I come along with you? |
| 00:45:09 | Then you have to come right now. |
| 00:45:20 | Never mind, I'll go, I also have to go to BEVO. |
| 00:45:23 | Well, then why don't you go instead? That makes a lot more sense, doesn't it? |
| 00:45:34 | Ladies? Ladies? |
| 00:45:40 | Well, go ahead then. |
| 00:45:47 | Where is (inaudible)? |
| 00:45:48 | Just to the restroom. |
| 00:45:49 | In the rectangle one diagonal. |
| 00:46:01 | It is explained over here? |
| 00:46:03 | Well, that is explained with the surfaces- |
| 00:46:05 | Yes. |
| 00:46:06 | How that works. |
| 00:46:07 | Oh, that is right here- |
| 00:46:08 | So you can go right ahead- and now you just have to calculate this again. Yes. |
| 00:46:10 | Oh, with the surface area? |
| 00:46:11 | Yes, because as far as the (inaudible) diagram is concerned that doesn't get explained till over here in exercise seven. |
| 00:46:15 | That's why you don't have to do that one. |
| 00:46:16 | And then over here it is explained again and then you can continue on working. |
| 00:46:20 | Plus five is 17- no, |
| 00:46:23 | It is 14. |
| 00:46:27 | :00] |
| 00:46:36 | Myrte? It was, uh, chapter one, wasn't it? |
| 00:46:38 | Yes. |
| 00:46:39 | Sorry, that one is wrong. |
| 00:46:43 | This is also wrong. |
| 00:46:45 | No it's not. |
| 00:46:46 | Yes it is. |
| 00:46:50 | Don't make me cry. |
| 00:46:52 | Is it that bad? |
| 00:46:58 | :00] |
| 00:47:18 | Check your answer by measuring this in your diagram. |
| 00:47:27 | The numbers on your triangle have completely faded. |
| 00:47:43 | Yes? |
| 00:47:44 | (inaudible) do wrong? |
| 00:47:45 | Excuse me? |
| 00:47:46 | I don't see why this is wrong here. |
| 00:47:49 | What is done wrong here? |
| 00:47:50 | Look it (inaudible) is wrong but how are you supposed to know? |
| 00:47:56 | What does that triangle look like? What kind of triangle is this? Why don't you make a diagram of that triangle. |
| 00:48:01 | One like that. |
| 00:48:03 | Yes, but with the sides and all, so do it with the correct- oh, you have it already. Oh, no, that is not it. |
| 00:48:08 | Although? Yes that's the one though. What does it say? Angle K is 90 degrees. Correct. K L is 15. |
| 00:48:15 | That is also correct. And then it says: L M is 25. And where did you put it? Yes? |
| 00:48:21 | Oh, here- |
| 00:48:22 | You see? And that is what has gone wrong here too. Do you see that? |
| 00:48:26 | Because first you had to take a right-angled side and then another right-angled side. |
| 00:48:31 | That is why it is so important that you really make the right diagram with the letters in the correct place. So that you know exactly what belongs with what. |
| 00:48:39 | Yes? |
| 00:48:44 | Ladies and gentlemen, it is almost time, so you can start to pack. |
| 00:48:49 | Yes? |
| 00:48:52 | Is it- if you, uh, write it down like this because you have to measure the other right-angled side- |
| 00:49:00 | Yes? |
| 00:49:01 | Then is this, uh, correct then, because, uh- |
| 00:49:03 | The calculation is correct but you still have to answer it. |
| 00:49:06 | Because now we have to search, no, you don't have to explain anything. |
| 00:49:10 | Because now I have to search for your answer. Like do I need 41 or do I need 40, or do I need nine? |
| 00:49:15 | Yes? |
| 00:49:16 | I need to know what you calculated. Then you can just say the other side is- well, what did you calculate? Forty. |
| 00:49:21 | Yup. |
| 00:49:22 | Yes? So you still have to give an answer because otherwise I will have to search in such a diagram. |
| 00:49:25 | Yup. |
| 00:49:26 | And you know that I will always find the wrong one. |