# NL4 EQUATIONS

This eighth grade mathematics lesson focuses on factoring. The lesson is 45 minutes in duration. There are 22 students in the class.

Time | Caption |
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00:01:35 | What kind of guy is that? |

00:01:49 | I will tell you that now, I will tell you that now. |

00:02:01 | Lis- listen for a second. |

00:02:12 | If you listen for a second, then I will tell you exactly. You guys see Ruud Gort, that is the name of this gentleman. |

00:02:20 | He will be taping a complete lesson on video, as part of a study that is held in various countries. Yes, that is what those videotapes are used for. |

00:02:40 | It is a study about math education in the Netherlands, compared to Australia and etc., etc. |

00:02:48 | He thinks it is best, and the recording works best if you pretend he is not here, okay? So we are going to act normal, as usual. What? |

00:03:00 | (inaudible) |

00:03:03 | (inaudible) |

00:03:04 | Aren't you? Or still just in time, just in time. |

00:03:05 | You guys are so late! |

00:03:10 | (inaudible) for the camera. |

00:03:19 | Oh I see. You wanted to step into the picture very explicitly. |

00:03:30 | Sir? |

00:03:31 | I was just saying... I will briefly repeat it once more. |

00:03:35 | That gentleman will be taping a complete lesson on video as part of an international study. That's why. |

00:03:46 | If you guys could already take your homework out in front of you. |

00:03:50 | And homework, according to the study calendar, was the last- the last part of the basic materials. |

00:03:54 | [ Bell ] |

00:04:02 | Sir! |

00:04:03 | The very last part. Bram? |

00:04:07 | I, uh, brought the wrong notebook with me. |

00:04:10 | That, that is very clever. I wanted you to demonstrate later how you do those types of exercises. |

00:04:18 | No! But I do have (inaudible). |

00:04:25 | Then, what have you just been doing during Dalton hour? |

00:04:31 | But that wasn't the homework assignment, was it? |

00:04:35 | Marian, what is it? |

00:04:36 | I don't understand 31. |

00:04:37 | Yes, well first you are going to... let's have a quick look... |

00:04:44 | Will you guys calm down a bit! Can you guys calm down a bit here? |

00:04:49 | Yes, but (inaudible) isn't here yet. |

00:04:51 | But you can still calm down a bit, right? That has nothing to do with it? I am sure he will arrive shortly. Keep that bunch quiet over there. |

00:04:52 | Okay, I'll keep those little ones quiet. |

00:04:56 | Very good. |

00:05:06 | As usual you can first ask questions yourself about the homework, what didn't go well. I will have, uh, one of your fellow students explain it. |

00:05:17 | And after that I will have a couple of questions myself about the calculations that you have learned in your homework. Marian? |

00:05:27 | Which assignment? |

00:05:28 | Thirty-seven. |

00:05:29 | The very last one. |

00:05:30 | Yes. (inaudible) |

00:05:31 | The very last one, I'll keep that in mind. |

00:05:34 | Twenty-eight C? |

00:05:37 | Twenty-eight C. Oh that was last time's homework, that's the previous homework. You are very late with that, aren't you? |

00:05:49 | Yes. |

00:05:50 | Why don't you take care of that during a Dalton hour. |

00:05:56 | Thirty-six. |

00:06:00 | Do I understand it correctly that this is only about the last two assignments? |

00:06:07 | :00] |

00:06:26 | The last two assignments, Sam help me out here. |

00:06:31 | Thirty-six? |

00:06:32 | Thirty-six, we will start with that one. Copy the table and fill it in. I'll start off with that and then you can tell me what else I need to write down. |

00:06:52 | What is it? |

00:06:53 | Didn't I have to stay behind now? |

00:06:55 | Can you guys just sit here and work? You can't just interrupt a class, can you? |

00:07:12 | The intention is that this equation gets resolved. |

00:07:21 | Right? So you have to put the right parts of this equation in the right place of this table. |

00:07:27 | Sander! And as a suggestion- as a suggestion the factor X is put up front already. How can you get this equa- multiplication to resolve? |

00:07:40 | The four X, uh, squared two divided by X (inaudible) four (inaudible). |

00:07:49 | Write down what you think is the correct answer and check it right away. X times four X, is that four X squared, or not? |

00:07:58 | Yes. |

00:07:59 | Yes, then it is correct. Here you do the same. X times what? |

00:08:06 | Twelve. |

00:08:07 | Twelve. And then you can write the equations with brackets. Hans? |

00:08:16 | Y equals X and between brackets four X plus twelve. |

00:08:23 | Exactly right. We are going to do that again in question C. This was question B, right? We are going to do it one more time. Daan? |

00:08:42 | We can fill in the terms of the equation in the correct places of the table. Daan? He wanted to put four X up front. |

00:09:01 | Resolve the equation, complete it. |

00:09:05 | Four X times- What makes four X squared? |

00:09:10 | Four X times X. |

00:09:12 | X. And four X times what makes 12 X? |

00:09:19 | Plus three. |

00:09:20 | Plus three. If you apply that, then you can write down- |

00:09:27 | Four X and between brackets X plus three. |

00:09:36 | You have to remember this carefully because this was a good exercise. |

00:09:40 | Remember carefully, if you have to resolve this equation, you can make it into this equation or this one. |

00:09:50 | They basically come down to the same thing, but the last one is better. And use that as a guideline. |

00:09:58 | You have to keep as much as possible in front of- in front of those brackets. |

00:10:06 | As much as possible. If you have to do this for a test, writing this equation between brackets, |

00:10:15 | then I won't consider this completely wrong of course, because in essence it is a correct equation. But this one is better. |

00:10:22 | So, this one will get you more points, than this one. Try to get as much out of it as possible. |

00:10:30 | You can also say, if I resolved- because this is called "to resolve", if you write the equation with brackets as a multiplication- |

00:10:42 | if I resolved then I always check if those two terms, that are written between brackets, still have a common divisor, a common factor. |

00:10:55 | And then you come to the conclusion that there is still a common factor four in there, we have to take it out as well. |

00:11:05 | That's the idea. Make sure you remember this. |

00:11:09 | We are not going to do every assignment of question 37, because that one was also requested. Not all of them, but a couple of examples. |

00:11:28 | I think that question B is actually a- a good example. |

00:11:48 | Tim? |

00:11:50 | Yes. |

00:11:54 | Where do I have to fill in these two parts? |

00:12:02 | Below the, uh... |

00:12:04 | Why don't you- just point it out in your notebook, then I know- What, no tables? Wait a minute. |

00:12:08 | No, I tried without any. |

00:12:09 | What have you got? |

00:12:14 | You have only the final results. Right? |

00:12:19 | Yes. |

00:12:21 | Did you do it yourself, without using a table? |

00:12:24 | I did it with the help of (inaudible). |

00:12:27 | Not with the help of the answer book, I hope? |

00:12:29 | Some of them I did. |

00:12:30 | Well, that's no use. That's the- the wrong attitude, Tim. |

00:12:34 | If you say, I can't do it without a table that's not a disgrace. Most of us can't and that's fine with me. But then make the table! |

00:12:49 | What do you learn if you copy it from the answer book? |

00:12:52 | Not much. |

00:12:53 | Well, you might as well say nothing at all. And your goal, I would hope, is that you learn something new. |

00:13:01 | Table, participate, right now, in your notebook. Right away, do the table. |

00:13:16 | Lisa, where do I fill this in? |

00:13:20 | You didn't finish it you said, but that doesn't matter. We're making it right here and now. Where does it belong? |

00:13:34 | Um, below the, below the multiplication, um, sign? |

00:13:37 | Did you draw the table? |

00:13:38 | Yes. |

00:13:39 | Point it out to me where- no, no, no, no, no. |

00:13:42 | Because that is part of a multiplication, in the equation it doesn't say that you have to multiply two X squared by something. |

00:13:53 | Oh, no you need to write it here. |

00:13:54 | There, yes. So the equation without brackets comes from these places in the table. Filled out. |

00:14:06 | Furthermore the book suggests that you fill in minus two X over here. |

00:14:16 | That was already printed, suggested. You only have to figure out this part. Lisa, please complete the multiplication correctly. |

00:14:27 | Minus two X times what is minus two X squared? |

00:14:33 | Um, I don't know. |

00:14:38 | Where did the little squared come from, Lisa? |

00:14:40 | Oh, times X. |

00:14:42 | Times? |

00:14:43 | X. |

00:14:44 | X, that's correct. Minus two X times what? Makes minus 54 X? |

00:14:57 | Twenty-seven? 27. |

00:15:05 | Exactly right, minus two X times 27 is minus 54 X. No flaw in that argument. Correct? So what equation do you get with brackets? |

00:15:17 | Minus two X, um, times X plus 27. |

00:15:27 | This is how you score an A for your test. And all those- since naturally I can't guarantee anything, |

00:15:37 | all those who looked at the answer book too often during homework won't be able to do this during a test. |

00:15:48 | And they won't just score a C or C+, but a D or even an F. |

00:15:56 | Because this is a subject in which you either perfectly understand what is going on, you kind of know what you are supposed to do, |

00:16:03 | regardless a little calculation error or a sloppy error, |

00:16:08 | or you don't understand the intention at all and then you can't do one single exercise. |

00:16:19 | I want to do one more of 37. And then we are going to have a short quiz on this lesson. Well let us do the very last one. |

00:16:36 | :00] |

00:16:59 | This is printed in full. That is what we are trying to work towards. |

00:17:04 | Well, it is in fact kind of a stupid remark, "that's what we are trying to work towards" because there is actually nothing printed. |

00:17:10 | The only thing that is printed is that you have to make something with brackets. Pleun, tell me, tell me, tell me... |

00:17:20 | Where does the minus three S squared come from in the table? |

00:17:21 | Um. Um, in front it says three S... |

00:17:29 | No, that is not my question, Pleun. This, where do I have to fill this in? |

00:17:33 | Uh, underneath. |

00:17:34 | Where? |

00:17:35 | Uh, underneath. |

00:17:37 | Yes there, that's right. And the plus nine S? |

00:17:42 | Well, next to it. |

00:17:43 | It goes next to it. You can leave out the "plus" or you can write it down there as well. |

00:17:50 | Yes, now you have to solve the puzzle. I do want to end up with a "plus" here. |

00:17:58 | Yes? Let's take a look. Suggest something. |

00:18:03 | Um... in front three S. |

00:18:05 | In front three S. |

00:18:08 | And then an S at the top. |

00:18:10 | And S over here. Correct, or not? Three S times S makes? |

00:18:13 | Um... no, I mean negative S. |

00:18:19 | Negative S. So, always check right away. As soon as you write something down, check it. And here? |

00:18:28 | Um, three. |

00:18:30 | Three? |

00:18:31 | Yes, three. |

00:18:32 | That's all? |

00:18:34 | Yes, add the plus. |

00:18:36 | Yes, you can add the "plus", but it is not necessary. Okay, equation? |

00:18:42 | That becomes three S and in between brackets minus S plus three. |

00:18:54 | Perfect, completely correct, right. Marian? |

00:18:59 | Where did you come up with three S and that minus? |

00:19:06 | What you can do, what you can do is for yourself, on a scratch note, or on top, resolve this completely to a multiplication. |

00:19:24 | Minus three S squared, what exactly does that mean? Minus three times... |

00:19:31 | S. |

00:19:32 | S, times- |

00:19:34 | Squared. |

00:19:35 | Yes, squared, but I want to resolve it completely to a multiplication. |

00:19:39 | S times S. |

00:19:44 | And nine S, completely resolve to a multiplication. Nine S is in fact... |

00:19:53 | Nine times S. |

00:19:54 | Nine times S. And the nine you can resolve even further. Write further as a multiplication. |

00:20:02 | Three times (inaudible). |

00:20:04 | Three times? |

00:20:05 | Three times S. |

00:20:06 | Three times S. And what do those two have in common? |

00:20:23 | (inaudible) last times S (inaudible). (inaudible) last times S (inaudible). |

00:20:30 | They both have a factor S in common. And what else? |

00:20:38 | A factor... |

00:20:41 | Three. |

00:20:42 | Three. This has a three in it and this has a three in it. And that is the common part, which you can write down in front. |

00:20:51 | Sir? |

00:20:52 | You also see now what is left over. |

00:20:58 | And that is written above here. Because here you only have a three left over. Right, so you can try to do this by heart. |

00:21:09 | But if you can't do it, then take a piece of scratch paper and write it all out on it. |

00:21:15 | I put, uh, the minus in front of the three S instead of the S. |

00:21:19 | Yeah, there's no space for that. |

00:21:21 | Okay, we will just do it, will just do it. That's this equation. |

00:21:39 | This is how we started. You wanted to do this. What will the upper part look like, Freek? |

00:21:53 | The S on top. |

00:21:55 | This is correct, yes. |

00:21:57 | And on the left, uh, the minus three. |

00:22:00 | Here? |

00:22:01 | Yes, (inaudible). |

00:22:03 | What equation would you write down next? |

00:22:06 | H is minus three S and then in brackets S minus three. |

00:22:17 | Uh huh. Freek, let me tell you something, son. If you do this on a test, that is fine with me. |

00:22:29 | The only reason why you actually shouldn't have done that in this particular exercise, is because that little "plus" sign is the only thing that was given in print. |

00:22:40 | And this way you will not get this. |

00:22:44 | I think personally, but that is my personal opinion, I think that this resolution is even- even better. But I would- |

00:22:51 | But is it correct then? |

00:22:52 | But I would, on a test, consider them both correct. I would not pre-print that information on the test. |

00:23:02 | You see? So I think that is a good solution and this one too. Peter? |

00:23:06 | But you have to end up with the highest number up front, in front, eh? |

00:23:09 | As many as possible. |

00:23:12 | Oh, as many numbers as possible or, letters and such? |

00:23:15 | Yes. As many as possible. As long as you can still see a common factor here then you haven't taken out enough yet. |

00:23:26 | Oh, I see. |

00:23:42 | Um, I am going to ask you to do an assignment that none of you have done yet. Just do that one. Page 145. Assignment E. |

00:24:00 | And just start, uh, first on E 12, start on E 12. Maximum five short minutes. |

00:24:11 | (inaudible)? |

00:24:12 | E 12. E 12, E 12 A, B, C. Maximum five short minutes. |

00:24:22 | Then I will ask a couple of calculations, that way we can check if you can apply the trick. |

00:24:34 | (inaudible) |

00:24:35 | If you need a table, then you'll have to make one. |

00:24:40 | If you are so good at this that you don't need a table, you may do that as well. But not with the answer book. |

00:24:59 | Then I will just have a look at who is missing. Five, 13, 14, 16, 18, 20, 22. I am missing- what, four people? (inaudible) Seventeen, 18, 20. |

00:25:19 | :00] |

00:25:40 | What happened to Patty? |

00:25:42 | Over here. |

00:25:43 | You went and sat over there? Okay. |

00:25:48 | :00] |

00:26:37 | Were is Christianne? Is she ill? |

00:26:43 | :00] |

00:27:25 | Let me see, let me see. That is correct, very well. |

00:27:37 | Sir? Sir? Is this a little plus sign? For if we put minus eight H over here this one over here is 24- |

00:27:44 | Uh huh. |

00:27:45 | Then again minus 24 H (inaudible) and then simply plus three? |

00:27:51 | This is right this way, isn't it? |

00:27:57 | Is it not working out? Let me see. Oh, that's going well, that's fine. |

00:28:04 | Though you didn't copy exactly what was shown in the equation, right? There is a little sloppiness there. |

00:28:14 | You had written down in the equation 18 X and? |

00:28:17 | Yes. Oh, minus. |

00:28:20 | Minus, you're missing the minus. |

00:28:28 | This one you can already put in the right place. |

00:28:32 | (inaudible)? Here? |

00:28:35 | No, just in front. |

00:28:38 | Here? |

00:28:39 | Yes there. You have to- to- to remove that. Yes, like that. Now that 18 isn't correct any more, is it? |

00:28:48 | No. |

00:29:02 | What? What? Because this you don't know yet. What times three gives you this? |

00:29:15 | That's right, like that. |

00:29:24 | Without a table. Let's have a look. Yes, that's going fine, that's going fine. |

00:29:38 | You should try to write an S differently than a five, don't you think? |

00:29:43 | Is this, is this an S then? |

00:29:44 | Yes, because I almost don't see a difference here. I am inclined to say it is 55. |

00:29:49 | Okay. |

00:29:50 | Try to differentiate them a bit. Yes? |

00:29:55 | Sir, I just don't feel well. |

00:29:59 | The second one. |

00:30:00 | But I, I just don't feel well. |

00:30:01 | What's the matter? |

00:30:03 | I just don't feel well. |

00:30:05 | Too sick to stay inside? Do you want to go home, is that what you mean? |

00:30:09 | No, because this is the last hour. |

00:30:14 | What do you want to do? |

00:30:16 | Outside the classroom, I don't really know. |

00:30:19 | Em... |

00:30:20 | I am just very nauseous. |

00:30:22 | All right. Just go together, to the bathroom, there you can drink a little water. The two of you go. Who should come along? Yes? |

00:30:38 | Yes? |

00:30:39 | With the other ones you could always fill in- there you could already do one- |

00:30:45 | The last one you can fill in something else, right? You placed this correctly. This one, where can you put it as well? |

00:30:54 | Here? |

00:30:55 | Uh huh. And now you can start. Resolve this multiplication. Yes? And now you can start with the second part. |

00:31:09 | Resolve that multiplication as well. That's incorrect, isn't it? Because then you get five. |

00:31:21 | (inaudible) |

00:31:22 | Yes, now it's still not right. Five times seven is not minus 37. Yes? |

00:31:29 | And then, in addition to that, S times S squared. That's not S squared. That would be S cubed. |

00:31:40 | That has to be S. |

00:31:41 | Yes, look at that, three mistakes in one cell. |

00:31:46 | But, fine, now it's complete and now you can write the correct equation down. Five minutes have been up for a long time. |

00:31:52 | Shoot, I'm only at three. |

00:31:53 | Well, you didn't have to finish it completely. The idea is that you know how this works. |

00:32:02 | Take your answer book in front of you then we can check if you did it right. Just whatever you, whatever you finished, you check. |

00:32:17 | Give me a signal if you came up with something different. |

00:32:23 | :00] |

00:32:44 | Bram. |

00:32:45 | Look at this one, it says minus, no, so that becomes three T. Minus three T, minus two squared. |

00:32:52 | Yes, yes? |

00:32:53 | B-6- |

00:32:54 | And then... |

00:32:58 | B-6. And then here I have, since minus three T- you've got to have a T here, in any case, and over there as well. |

00:33:07 | Yes, well, your resolution is correct; your resolution is also a good example of what I wrote on the center board. I just told you: you could have gotten more out of it. |

00:33:18 | Because those two- do those two still have anything in common? |

00:33:22 | Oh, I could have made this three T. |

00:33:25 | Yes. Yes, you are better off making a new table because you are crossing out so much now. |

00:33:47 | So, here just a minus? |

00:33:49 | Yes, but you also need to add a number, right? You will never multiply with just signs, will you? Yes, like that. Yes, that's correct. |

00:34:05 | There. |

00:34:06 | Yes, which resolution are you doing in that case? |

00:34:10 | And now here- here you took out minus T. Where is- |

00:34:14 | It is now over here. |

00:34:15 | It is now over here. While in the assignment it is suggested that you end up with a plus. You could have made sure of that by taking out the minus. |

00:34:27 | Yes, then you have to remove it there. |

00:34:33 | Over here nothing. |

00:34:34 | And remove that one as well- |

00:34:35 | So then this becomes- |

00:34:36 | And now you can make the correct equation. |

00:34:43 | Gives minus three T, one T. |

00:34:46 | Yes. Yes? |

00:34:50 | So, your equation wasn't wrong but you could have done it a little bit better. |

00:35:04 | Which one didn't work? |

00:35:05 | I had this one differently. |

00:35:07 | Well, that's a good example of what I wrote on the center board. Your resolution is correct, but you could have done it better. |

00:35:14 | The seven P and the 21, do they have anything in common? |

00:35:19 | Um. Yes. |

00:35:20 | What is it? |

00:35:21 | Um, let me see. Ah, the seven itself. |

00:35:26 | So the seven you could have also taken it to the front. Yes? And it has an effect on the top and that way it'll be all right. |

00:35:33 | So your equation isn't wrong but not exactly correct either. |

00:35:40 | Let me put it this way, Lisa, if I would give, for example, four points for this assignment on a test, then you would still have gotten three. |

00:35:41 | Oh, right. |

00:35:48 | Because the resolution is fine, but it could have been a bit better. Someone who takes out the seven just deserves that extra little point. |

00:35:56 | Yes. Can I take your eraser? |

00:36:03 | Minus three T and then why is this here- |

00:36:06 | Wait a second, wait a second. Um, for those of you who did it right and finished the checking by now, you can continue working. |

00:36:18 | Homework for next time is given in your study calendar. |

00:36:22 | G and P. |

00:36:23 | The G and the P, but you will never be able to finish it all, because that would be four pages. |

00:36:27 | The agreement is always at least two pages of homework. You can choose a few of the G assignments and a few of the P assignments, |

00:36:37 | because you guys have to know the P section as well. Okay? |

00:36:42 | Here it says minus three T. Do you have to fill that in over here? |

00:36:46 | Wait a second, wait a second. Assignment D. Let me see... you can fill this in the right place, which is not the problem. |

00:36:53 | Yes. And then- |

00:36:54 | And now you have to solve the puzzle. |

00:36:55 | Yes. |

00:36:56 | And you solved the puzzle. This is completely correct. This is a good example of what I did earlier on the middle of the board. Your equation is correct. |

00:37:06 | But you could have done- |

00:37:07 | Oh- |

00:37:08 | You could have done it better. Because do those two still have anything in common? |

00:37:10 | Oh, yes. Well, yes. |

00:37:12 | Yes. |

00:37:13 | The factor. |

00:37:14 | Which one? |

00:37:15 | Three. |

00:37:16 | You could have taken three out as well. |

00:37:17 | Oh, right. |

00:37:18 | Furthermore, it says here, not that that's a very big deal, here it said a plus and you wrote a minus. |

00:37:25 | Oh yeah, that's right. |

00:37:26 | So, if you also put the minus in front, you better make a new table- |

00:37:30 | Oh right. |

00:37:31 | That way you won't take out a T but? |

00:37:36 | Here you don't write a T but? |

00:37:38 | Oh, but minus three T? |

00:37:39 | Yes, and if you complete it, then it will work itself out. Just make a new table, okay? |

00:37:45 | This one, wait a minute over here. I had, this one is here, and then I had this one- |

00:37:54 | Yes, that is the same story, actually. You have two H and a six. Do these two still have anything in common? |

00:38:03 | Yes. |

00:38:04 | What? |

00:38:05 | Yes, you can divide them again by- oh no- |

00:38:08 | Yes, though. |

00:38:09 | Yes that, by three uh... |

00:38:11 | No, two H and six you can both be divide by? |

00:38:14 | Two. |

00:38:15 | Two. And this two you could have taken out as well. |

00:38:16 | Oh. |

00:38:19 | You are better off in making a new table, otherwise you may not understand it anymore. |

00:38:22 | And you don't need to write the one down, do you, at uh... |

00:38:23 | No, you don't have to write it down, but you do have to write the first part of the equation down. |

00:38:29 | Oh. |

00:38:30 | That's part of it. |

00:38:31 | Okay. |

00:38:32 | It is a small thing, but you have to do it. Okay? Just make a new table, |

00:38:35 | but then don't write four H but eight, minus eight H, that will make it correct. |

00:38:48 | Sir, I don't get- do I need to finish C? |

00:38:49 | Yes. |

00:38:50 | Huh? |

00:38:51 | I understand the equation, but I don't know how to explain it. |

00:38:52 | Why? |

00:38:53 | I do understand that I have to end up with six plus nine and one, (inaudible). |

00:39:00 | Just look carefully. Um, do you still remember that Australian gentleman? |

00:39:01 | Try to find the patterns, patterns. How can you determine the little blocks from the first figure? That can be done in many ways, |

00:39:15 | One could say I do three times four. Or uh, sorry three times three. Three times four, three times five. Etcetera. |

00:39:22 | But you have to make a combination with that number. |

00:39:27 | The other thing you can do is, let's say, I am going to take six cells plus one times three. Here, I will take the same six cells plus two times three. |

00:39:43 | Oh, right. |

00:39:46 | And over here six cells plus three times three. Etc. Then you can use this equation. |

00:39:52 | Okay. |

00:39:53 | It depends upon what pattern you see, because you may see a slightly different pattern there then Daan does. |

00:39:58 | Which would mean that if you have to make your own equation, then you could find the equation in a slightly different form than Daan. |

00:40:05 | But they can both still be correct. |

00:40:07 | Okay. |

00:40:08 | But you have to find the connection with the number that is given with it. That's your aim. |

00:40:14 | If you succeed with that then you can immediately calculate the amount for every given number. |

00:40:22 | But if all you see is this pattern: 9, 12, 15, 18, etc., can you tell immediately? Uh, what is in number 30? |

00:40:32 | Ninety-six. |

00:40:33 | Without writing down the whole list. Right, so- |

00:40:37 | Ninety-six (inaudible) should be correct. |

00:40:39 | Yes, maybe, but how do you explain that? |

00:40:41 | Well, I did it like this, I thought three of these little blocks make one row, that left me with two rows, that means six. |

00:40:48 | And then I had 30 left; 30 times three plus six. |

00:40:52 | So you are actually working with the equation that is shown there. Did you cheat a little? Or did you find it yourself? |

00:41:00 | Yes, I thought it sounded logical, actually. |

00:41:02 | Yes, well, that is in fact the trick of these types of exercises. Can you recognize a pattern and then try to involve that number in it as well. Right? |

00:41:13 | Look, if all you say is three are added each time. |

00:41:17 | So you have to keep adding until you're there? |

00:41:18 | That will get you there as well. Sure, that will get you there as well. That way you can say three are added each time. |

00:41:23 | I just asked for number 30. This is number one. |

00:41:28 | But you do have to indicate the step. After number 30 there are another 29 left. And if you argue from there on you will figure it out as well. |

00:41:37 | Sir! |

00:41:38 | Yes? Try to keep making a connection with the numbers up above. |

00:41:45 | You will be known all over the whole world, right, that this is your math notebook. They are going to be looking at it in Hong Kong. |

00:41:54 | Then they will know right away that they may take a piece of paper of mine. |

00:42:01 | Tell me, what did you want to ask? |

00:42:04 | The T. Why is the T R? |

00:42:07 | Yes? |

00:42:09 | I just see- how, how, how do you have to do that? |

00:42:12 | Can you do T one, T two, T three, T four, T five, T 20. Here, you had to, to demonstrate two of them- |

00:42:20 | Yes I've got those. |

00:42:21 | Oh, you've got those? So at T five you calculate 10 plus four, times three. |

00:42:27 | Yes. |

00:42:28 | What has that got to do with this five? |

00:42:33 | What is the relationship between the number of the term and your equation? |

00:42:37 | It is one less each time. This one is... |

00:42:40 | Is that correct here? |

00:42:41 | Yes. |

00:42:42 | At T 20? |

00:42:43 | Nineteen. |

00:42:44 | Why don't you go ahead and apply that. |

00:42:45 | Oh right then that makes N minus one. |

00:42:47 | If you write nothing in between there then you mean times, right? You do have to write something in between there. Yes, like that. Well, perfect isn't it? |

00:42:55 | But how does it work here? In the row five, seven, nine, 11, 13 is T one five. Then this must be T two, T three, T four, T five- |

00:43:02 | Yes. |

00:43:03 | The difference between two terms is the same each time. We call a series like that an arithmetical series. |

00:43:08 | Yes. |

00:43:09 | Each number in this series is T one plus N, minus one. |

00:43:15 | Times two. |

00:43:16 | Yes. |

00:43:17 | It is almost the same as the one you got here, right? |

00:43:22 | But then you explain why the equation for the arithmetical series can be written as T N... T and... |

00:43:33 | I don't understand that one. |

00:43:34 | Do you recognize the part that is on the board. |

00:43:38 | Yes. |

00:43:39 | In this, in this equation? |

00:43:40 | Yes. Those brackets. |

00:43:46 | This is a multiplication. Can't you write that one without brackets? |

00:43:55 | Go ahead and try, let's say try to get rid of those brackets entirely. |

00:44:04 | X- But may I now write this one here as well? |

00:44:09 | Yes, since the sequence is irrelevant in a multiplication. Sure, go ahead. |

00:44:17 | Don't write two to the power of N, Pleun. You haven't gotten there yet, have you? |

00:44:23 | It wasn't N squared either, was it? What is two times N? |

00:44:29 | Two N. |

00:44:30 | Not two to the power of N, but just two N. That's right. |

00:44:47 | Where did you get this three? |

00:44:48 | Yes, well just copy the equation again. What, what equation do you end up with? |

00:44:53 | It, it is this one, right? T N- yes, write it down, T N equals... equals? |

00:45:04 | T one- |

00:45:06 | One plus whatever is in your table. |

00:45:07 | Two N minus two. No without "times two" then. |

00:45:12 | No, no, no this is correct. Now you can figure out what T one was. What was T one? In this- this row? |

00:45:27 | Five. |

00:45:29 | Okay, fill it in. Write the equation down again. |

00:45:38 | And try to shorten it, Bram. |

00:45:46 | I don't get B three. What are you supposed to do here, do you have to simply- |

00:45:49 | Yes, fill in the equation. |

00:45:50 | Do you simply have to do one plus three? |

00:45:51 | Yes, this means that you fill in one for N and that means that you have to fill in twenty for N. |

00:45:59 | So you have to do twenty plus three? |

00:46:02 | Yes, so you only have to fill it in into the equation. |

00:46:04 | Twenty times 20, equals 20 squared. |

00:46:05 | Huh? |

00:46:06 | And so here you put 20 squared. |

00:46:08 | Yes, and then you have to calculate it further. Yes? |

00:46:15 | Isn't it better here to calculate minus four T plus three? |

00:46:17 | Yes, yes, yes, your table is not completely correct. The table is not really correct because minus four T times two- |

00:46:26 | Oh, no that's not possible, is it? |

00:46:27 | You will never get eight T squared out of that. |

00:46:29 | Minus two, right? |

00:46:31 | That for sure. Wait a second, it still doesn't make sense. Minus four T times minus two, if you don't look at this. |

00:46:41 | If you calculate this multiplication what do you get then? |

00:46:45 | Um, eight T. |

00:46:47 | And here, you have eight T squared. That's correct. Just- |

00:46:52 | Yes, now it should work. |

00:46:53 | Just continue changing the rest. |

00:46:55 | What? |

00:46:56 | Just change the rest of your equation. Yes, perfect. Just a minor error, with big consequences. |

00:47:09 | :00] |

00:47:28 | Well, it is almost time. You guys know- I didn't say that, Freek! You know what you have to do for tomorrow. |

00:47:37 | At least two pages, more is optional, more is optional. |

00:47:44 | G and P. |

00:47:57 | Can I reserve the fifth for tomorrow? |

00:48:07 | Tomorrow the fif- Yes, I think that will work out. You just have to write it down in the small folder, this, uh, that little red folder. |

00:48:24 | Hey, Tim, that doesn't lead to anything, right? Copying the answer book. |

00:48:34 | You will flunk the test that way, really. But I am actually happy that you admit it now. |

00:48:47 | Yes, but (inaudible). |

00:48:52 | Well, it is quite simple, if you only use equations- and you may do it without a table, then of course you can copy it from the answer book. |

00:48:54 | [ Bell ] |