NL4 EQUATIONS

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

Time	Caption
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 ]