# NL1 GRAPHING LINEAR EQUATIONS

This eighth grade mathematics lesson focuses on making linear formulas. The lesson was originally 46 minutes in duration; however due to the editing out of a video segment that was shown during the class, the footage in this version is 42 minutes. There are 24 students in the class.

Time | Caption |
---|---|

00:00:05 | But you did make it? |

00:00:06 | No. |

00:00:07 | Once again you forgot to? That's not too smart. Did you at least bring your books? |

00:00:14 | Yes. |

00:00:15 | Well, I'll make a note of it. |

00:00:21 | There's a couple of assignments that I didn't get back yet. |

00:00:23 | Which ones? |

00:00:24 | The- two of the additional exercises: two B, three and four. |

00:00:28 | I didn't even get to do any of the additional exercises, Miss. |

00:00:31 | Because I had a catch-up test (inaudible) I didn't have any time left to do this. |

00:00:33 | Yes. So you're at two now? |

00:00:37 | Yes, two B. |

00:00:39 | Are you going to catch up with that before the next class? |

00:00:40 | Yes. |

00:00:41 | Yes? |

00:00:42 | I didn't do the additional exercise at all. |

00:00:43 | Because? |

00:00:45 | Because (inaudible). But I did do this. |

00:00:50 | This one you did make? |

00:00:51 | Yes. |

00:00:52 | You too, Arnela? |

00:00:53 | Eh- |

00:00:54 | Did it work out? |

00:00:55 | Well, except for the last one. |

00:00:56 | Except for the last one. |

00:00:59 | Very well, we are going to begin. |

00:01:04 | Will you take your books and notebooks out in front of you and bags on the ground. |

00:01:12 | Marian, will you also put your bag on the ground? |

00:01:19 | And then you may open up your books and notebooks. |

00:01:22 | We may? |

00:01:23 | You may. |

00:01:27 | We are, uh, first going to discuss two problems. Then we will watch a section of video on this chapter. |

00:01:36 | And I understand from several students that not all of the additional exercises have been done by everyone, |

00:01:41 | not to say that some haven't done them at all. |

00:01:44 | So, this will then be- |

00:01:45 | (inaudible) |

00:01:46 | Sorry? |

00:01:47 | Is that for Wednesday? |

00:01:49 | Yes (inaudible). |

00:01:50 | I think you- |

00:01:51 | (inaudible) |

00:01:52 | Yes, but not completely. Some people haven't finished that section completely. |

00:01:56 | Uh, so we will just add that to the homework for next time. Um, I won't discuss it for now. |

00:02:01 | But you can ask questions about it. Especially those people who had such difficulty with those, uh, uh, equations, |

00:02:09 | for that we'll have to do some hard work during this class session, yes? We will start with problem 23. |

00:02:10 | Take that one out in front of you. Twenty-three. |

00:02:18 | I'm starting (inaudible). |

00:02:21 | Charles! Very cozy how you chose to sit in the back like that. |

00:02:25 | That way I can see better. |

00:02:27 | Are you going to participate and would you sit up straight, though? |

00:02:33 | Problem 23, a graph has been drawn there, Brian, and shown is a- well, you have to make an equation with it. |

00:02:46 | And that is actually what we talked about the entire last class session, |

00:02:50 | so I'm curious to know whether you succeeded in making this type of equation alongside a graph. |

00:02:57 | I succeeded. |

00:02:58 | You succeeded? |

00:02:59 | Yes. |

00:03:00 | Beautiful. |

00:03:01 | Succeeded? |

00:03:02 | I didn't. |

00:03:04 | Succeeded even marvelously. Linda? |

00:03:07 | No way. |

00:03:08 | Make my day. |

00:03:09 | Yes, because you skipped a class, right? Marion, did you succeed? |

00:03:14 | Yes, a little bit. |

00:03:15 | A little bit? |

00:03:16 | Not exactly. |

00:03:18 | Succeeded? |

00:03:19 | No. |

00:03:20 | No? Too bad. |

00:03:24 | (inaudible) |

00:03:25 | Well, let's just have a look then. I am just going to write down some equations. Uh, your answers. |

00:03:31 | Um, Simon, what did you get? |

00:03:34 | Uh... |

00:03:35 | Nothing. |

00:03:37 | What is your answer to question 23 A? |

00:03:39 | Oh, I don't have that one. |

00:03:40 | You don't even have that one. Okay, Tonny? What is your answer? What's the equation? |

00:03:46 | Twenty-three. |

00:03:50 | Peter? |

00:03:51 | Y times zero point five plus one equals X. |

00:03:54 | Y times zero point five... |

00:03:55 | Plus one- |

00:03:56 | times zero point five, and then? |

00:04:00 | Plus one. |

00:04:01 | Plus one. And then? |

00:04:04 | Equals X. |

00:04:05 | Equals X. |

00:04:10 | Sharone? |

00:04:11 | Uh, Y is three plus one times X. |

00:04:17 | Oh, now- now you lost me. One plus? |

00:04:20 | A half, |

00:04:21 | A half, |

00:04:22 | Times X. |

00:04:23 | Times X. Okay, Stefan? |

00:04:28 | Y is a half times X plus one. |

00:04:32 | Okay, Mina? |

00:04:34 | I have that. |

00:04:35 | You have that one too. Any more different answers? What do you notice about these last two equations? |

00:04:41 | They are the same. |

00:04:42 | The other way around. |

00:04:43 | Yes, they are simply the same. It's fine if you first write the one down, or the half times X. So these are the same. |

00:04:50 | Usually we take the bottom one, that is the way we have practiced it lately, first with the X and then the separate number in the end. |

00:04:56 | But this one is also correct, that is allowed as well. And let's go to the top one. |

00:05:04 | Is that one different or is it also the same? Maarten? |

00:05:07 | The Y is reversed. |

00:05:08 | The X and the Y are reversed. You have "equals X" and they have "equals Y". Only they started with that, yes? |

00:05:18 | How do you know what letter, what letter is separate? What will the answer be? How can you see that in the diagram? |

00:05:27 | The vertical line. |

00:05:29 | That one is on the vertical axis, Peter. What you fill in is always located on the horizontal axis, that's the X and what is on the vertical axis- |

00:05:37 | you always do that secondly like we saw yesterday. That is the outcome and that has to be placed at the end. Yes? |

00:05:44 | Very well, so that is the equation. Uh, question B. Calculate the value of Y if X equals 10. |

00:05:49 | All you need to do there is to fill in a 10 in the place of the X. |

00:05:55 | Y equals- I will copy that one half, times 10 plus one. First do times, one half times 10 equals five. And next do five plus one, which equals six. |

00:06:08 | And so the answer is Y equals six. Okay, we will go to problem 27. |

00:06:18 | Problem 27 is actually similar to the one we did yesterday and that was tricky. |

00:06:24 | It is the last problem in, uh, the section, so we should really take a look at that one. |

00:06:28 | I want to ask while you're looking at this problem- |

00:06:34 | For which of course you'll have several answers because you have checked which one belonged where, to which line, |

00:06:39 | that you take one- when I call your name, you may select one- one equation and then you may explain to me why it belongs to that line. |

00:06:48 | Who wants to do the first one? Any volunteers? |

00:06:52 | (inaudible) |

00:06:54 | Arnela, would you like to try one? |

00:06:55 | Um... |

00:06:56 | Of course you do, come on. We practiced them yesterday. No, you may choose the equation. Which of those four equations would you like to do? |

00:07:05 | Let's do... um... A perhaps. |

00:07:10 | A perhaps. Uh, A. The equation of A is four minus three X equals Y. |

00:07:21 | Okay, try to explain to me which one of these four lines belongs to that. |

00:07:27 | Uh, well, personally I thought it might be one. |

00:07:29 | You thought one. Explain why. |

00:07:32 | Well, because it, uh, well, I thought that those black lines crossed the vertical line at four. |

00:07:44 | You thought that this intersecting point would be approximately at four. Yes? |

00:07:50 | Okay, but why, uh, what if the four were over here? Is that also still possible? |

00:08:00 | Who can explain a little bit more? This line is a descending line, can you also tell that from the equation? |

00:08:07 | (inaudible) descending value. |

00:08:09 | Yes. |

00:08:10 | And the one the least was subtracted from, that's the number one. |

00:08:15 | Okay. Which ones, uh, are descending? I will write all four of them down. Uh, we have this one. |

00:08:22 | We have Y equals two-and-a-half X minus three. We have Y equals two-thirds X, and we have Y equals five minus three X. |

00:08:35 | Mark, which of the four are descending? Descending lines? |

00:08:40 | Um, A and D. |

00:08:41 | A and D. A and D. How can you see that? Ricardo, how can you see if an equation belongs with a descending line? |

00:08:51 | It runs like this. |

00:08:52 | Yes, you can see it on the diagram. But how can you tell that from the equation? |

00:08:56 | If it has a little minus mark. |

00:08:57 | Yes, but where does the minus mark have to be? |

00:08:59 | It has to be, um, it has to be, um, in front of the slope. |

00:09:05 | In front of the slope, yes. Because I see three minuses here, and yet there are only three descending lines. |

00:09:12 | Yes, the slope, that is the number by which you multiply so this one is descending, |

00:09:19 | that one is not, that is just the starting number. And this one is descending. So, either this is one or that is one. |

00:09:28 | Let's take a look at the starting number. Douwe? What is the difference in starting numbers between these descending lines? |

00:09:36 | Are you still able to see that, or not? |

00:09:37 | Not really. |

00:09:38 | Not really. |

00:09:48 | It's a bit light isn't it? |

00:09:51 | Like this? |

00:09:54 | It is also shown in your book, right? |

00:09:58 | That may be easier. What is the starting number of A? |

00:10:04 | Um, four. |

00:10:06 | Four. What is the starting number of D? |

00:10:09 | D? |

00:10:10 | D. Dirk? |

00:10:13 | (inaudible) |

00:10:15 | No, that is not the starting number. |

00:10:18 | Five. Here, the starting number is four and there the starting number is five. Oh, then this must be four and then that must belong to the five. |

00:10:32 | Then this one will belong to one, and that one belongs to two. |

00:10:41 | I still have two equations left. Which one belongs where? Um, Arnold? |

00:10:49 | Uh, B is three and you add that (inaudible). |

00:10:57 | Starting number is minus three. Well, this is the only one that has starting number below zero. So most likely minus three lies here. |

00:11:05 | And this one has exactly zero as a starting number. How can I find that one in here? Niels? I don't see any zero here. |

00:11:16 | No (inaudible). |

00:11:22 | Charles? What is the deal with that? |

00:11:26 | Uh, one. |

00:11:27 | No, but I would like to have a word with you shortly, Charles. |

00:11:30 | That's fine. |

00:11:31 | Very well. Maarten? |

00:11:33 | You don't have to write the zero down. |

00:11:34 | Yes, it actually has plus zero behind it, but they didn't write that down. |

00:11:39 | They just left the zero out, plus zero you can just leave that out. |

00:11:42 | The starting number zero, and that is this line. Okay, uh, I'm noticing that for several people it hasn't quite sunk in. |

00:11:50 | So I am just going to walk around shortly during the additional exercises, and I'll check how things are going and what you wrote down. |

00:11:55 | There were several people who didn't have problem 23. I'll stop by and take a look at those, too. And, uh, then you need a little extra practice with those. |

00:12:02 | Yes? Okay. We are just going to have a look at a little piece of video about this chapter. |

00:12:09 | And while you're watching I want you to, uh, uh, try to remember from the video in which professions this subject- graphs, uh, relationships, |

00:12:21 | something might even show up about a slope and a starting number, |

00:12:25 | so in what professions you will come across that. I would like you to tell me those things afterwards. |

00:12:31 | :00] |

00:12:39 | :00] |

00:12:44 | :00] |

00:12:51 | :00] |

00:12:57 | :00] |

00:13:04 | :00] |

00:13:11 | :00] |

00:13:17 | :00] |

00:13:24 | :00] |

00:13:30 | :00] |

00:13:37 | :00] |

00:13:43 | :00] |

00:13:50 | :00] |

00:14:05 | :00] |

00:14:09 | :00] |

00:14:18 | :00] |

00:14:23 | :00] |

00:14:31 | Okay, here is where we stop. |

00:14:36 | You have, um, come across the linear relationships within several things. |

00:14:42 | Um, Casper, why don't you name one? |

00:14:44 | In scuba diving. |

00:14:45 | Excuse me? |

00:14:46 | In scuba diving. |

00:14:49 | In scuba diving, right. Have any of you ever dived? |

00:14:53 | Yes? How was that? Fun? |

00:14:57 | I always find the breathing so difficult under water, so eh, huh? But do you do it more often, or just once? |

00:15:03 | In The Netherlands or uh, somewhere? |

00:15:05 | In France. |

00:15:06 | Well, well. And you, Ricardo? |

00:15:08 | Well, soon I'm going to dive with my (inaudible) at a diving beach. |

00:15:12 | Oh, yes. Exciting! Jolly. And what else did we come across? |

00:15:16 | Cell phones. |

00:15:17 | With cell phones, right, when you want to compare prices. |

00:15:21 | Then you have a starting number, that is the starting amount that you pay for the subscription and how much per unit is added. |

00:15:27 | The braking distance. |

00:15:28 | Excuse me? |

00:15:29 | The braking distance. |

00:15:30 | The breaking distance. Was that linear? |

00:15:31 | No. |

00:15:33 | No, that was not linear, was it? The faster you drive, the more is added to braking distance. Yes, that was not a linear relationship. |

00:15:41 | The skating. |

00:15:42 | The skating. Yes, we looked at that. |

00:15:44 | Alcohol checks. |

00:15:45 | Alcohol checks. With alcohol it wasn't linear either, right? If you start drinking more, that won't be quite like in a linear graph, no. |

00:15:55 | Very well, anyway we came across quite a lot of different things. Um, I am going to assign your homework and then you can get to work. |

00:16:00 | I will just walk around, therefore raise your hand if you have any questions. That is for... tomorrow. |

00:16:12 | Can you pass me a sheet? |

00:16:17 | Fifth period... |

00:16:25 | Do page 53 and page 54. And in case you didn't complete page 52 yet- |

00:16:35 | That's an awful lot. |

00:16:36 | Yes. |

00:16:37 | Those are two pages, yes. |

00:16:38 | Yes. |

00:16:39 | Nothing out of the ordinary. |

00:16:41 | Then you still have to add that page, right? |

00:16:47 | I will be right with you. |

00:16:57 | :00] |

00:17:00 | Yes. |

00:17:01 | Uh, tomorrow I will have a look at notebooks again, "by the way"... |

00:17:04 | By the way... |

00:17:05 | By the way... So be sure to have that taken care of. Just a minor warning. If not, then I'll have a neat surprise. |

00:17:15 | Oh, we'd love a neat surprise. |

00:17:17 | That is what I thought. Is your notebook in order? Oh, what a nice looking folder. |

00:17:23 | I thought that you were- you were going to make an appointment with me yesterday? |

00:17:26 | What? |

00:17:27 | Yesterday you said "I'll swing by to make an appointment" because you would have your notebook, eh, |

00:17:33 | with you, remember that? But you were gone at the end of class. |

00:17:36 | A double appointment. |

00:17:37 | Why don't you just remain seated for a moment at the end of class, okay? |

00:17:39 | Yes. |

00:17:41 | And your notebook? |

00:17:43 | I've made all the assignments. |

00:17:45 | Yes, would you put that neatly into a folder? Yes? So we can check it. |

00:17:52 | Fifty-two, 53, 54. |

00:17:54 | Next time I want to check your notebook to see if everything is in there. |

00:17:59 | Teacher, may I (inaudible)? |

00:18:04 | How about just making it by yourself? How far are you? |

00:18:07 | At 26. |

00:18:08 | Sorry? |

00:18:09 | Twenty-six. |

00:18:10 | Then keep going with it. |

00:18:14 | I am already at the additional exercises. |

00:18:16 | Let's see for a second those exercises from, uh- yes, the ones you did. Check those for a moment. |

00:18:23 | And Charles, why where you not participating earlier? |

00:18:27 | Yes, but seems to occur quite a lot these days. Your homework is not in order. |

00:18:31 | No, I didn't understand it... |

00:18:33 | I am not happy about that. Is one of your parents attending the upcoming Parents Night? |

00:18:38 | Because I will have a talk about this. I have warned you so many times and now I want you to finally straighten up. |

00:18:44 | Will it be done tomorrow? |

00:18:45 | Yes. |

00:18:46 | Yes? |

00:18:47 | Yes, tomorrow it will be finished. |

00:18:48 | Excellent, so for now you better get on top of it. |

00:18:50 | Teacher? |

00:18:51 | Yes? |

00:18:52 | I would love to have my grade of my math paper. |

00:18:55 | I will have a look. Yes? And you are not sure yet either? |

00:19:00 | I don't understand problem 25. |

00:19:04 | Twenty-five. In the graph you will see how the amount of water goes down in the boiler during showering. |

00:19:11 | Write down a linear equation in which you can calculate how much- well... |

00:19:16 | I don't understand that. |

00:19:17 | So you should have? What do you need for a linear equation? |

00:19:19 | Oh, uh, the starting number and the slope. |

00:19:22 | Very good. What is the starting number? How do you find that in- |

00:19:25 | Two hundred, that's- |

00:19:26 | Well, very good! Okay, let's start with that. Well, you just had the letters the wrong way around. |

00:19:32 | Yes. |

00:19:33 | Which letter is the outcome? |

00:19:38 | Uh... Y. |

00:19:40 | Yes, that is always the Y. Which Y axis is here? Yes, very good. Yes, well, you have to pick a letter for that. |

00:19:46 | Uh, just choose a letter. Uh, what letter shall we pick for that? |

00:19:52 | The amount of water or something |

00:19:53 | Well, multiply it by one letter, just one letter. |

00:19:55 | H. |

00:19:56 | H. Okay, "H equals" is a good way to start. |

00:19:58 | H equals... |

00:20:00 | The slope- or, uh, the starting number always comes all the way at the end. |

00:20:05 | Yes. |

00:20:07 | So you can go ahead and write that down already. Yes, it is a positive 200. |

00:20:14 | Uh, I am under the impression that not everyone is working. |

00:20:18 | I can't really see why, but... go ahead and get started, I will be by in a minute. Come on. |

00:20:29 | So it is? |

00:20:30 | Positive. |

00:20:31 | Positive. So do you add it up or do you subtract it? |

00:20:34 | Add it up. |

00:20:35 | Well, then write a little plus in front of it. |

00:20:37 | Okay, now we need to do something with the other letter. What letter will we take for the other axis? |

00:20:42 | No, this will be- this will become one equation. |

00:20:45 | Oh- |

00:20:46 | Just like you have made one equation here. |

00:20:49 | Where do you have the other one? |

00:20:50 | Yes, it looks kind of sloppy. |

00:20:51 | Yes, that one you should write down a little more neatly because it's difficult to reconstruct this way. |

00:20:58 | Let me see, X- |

00:20:59 | Y equals- put that one up front. And then the number you multiplied with. That was zero point five times X, simply X. |

00:21:11 | Something like this. |

00:21:12 | Okay. Plus one, it says there. That is how this equation should be as well. You already have "H equals", that is the same as this. |

00:21:20 | H equals, and then the slope- |

00:21:22 | Here you have plus one and over here you have plus 200. Now you still have to- |

00:21:24 | Insert the slope. |

00:21:25 | Correct. |

00:21:26 | That is, let me see, that is minus, that is minus one- oh, that is minus- |

00:21:33 | It is minus anyhow, you've seen that correctly. Go ahead and start by writing the minus down, because it descends. |

00:21:37 | And the slope is always when you go one step to the right. But that's kind of hard to see, how much one step is. |

00:21:43 | Can you tell me what would happen if you went 10 steps to the right? How many steps do you have to go down, then? |

00:21:50 | Uh, hang on, also, also 10. |

00:21:53 | Let's just have a look. Go from 10 to the right, we will arrive at this point. Where are we? |

00:21:59 | At 100. At the hu- like this at the 100 one- four steps? |

00:22:09 | Do you have a pencil for me? |

00:22:15 | Come on, start working. |

00:22:21 | That's this point. |

00:22:23 | Oh, that point. |

00:22:24 | Here it lies at the 10, and here it lies at the? |

00:22:28 | One hundred twenty-five. |

00:22:30 | Okay, so if I go 10 steps to the right, then I have to go down 125. If I go 10, 10 steps to the right, |

00:22:45 | I have to go down 125. |

00:22:48 | But I would really like to know how that works with one step, with one step to the right. |

00:22:57 | That has gotten how many times smaller? |

00:22:58 | Ten. |

00:22:59 | Divided by 10. |

00:23:00 | One hundred twenty-five divided by 10 is twelve-and-a-half. |

00:23:02 | Okay, and this is the vector number, or the slope. |

00:23:07 | Oh... |

00:23:08 | And that was a negative- and you still need to put a letter in front of the axis. Did you also have a question? |

00:23:13 | Yes. I, um, I was sick yesterday so I don't have 12 through 19. |

00:23:21 | Yes, okay. And did it work out? |

00:23:24 | Well no, I still need to do that. |

00:23:26 | Those ones you haven't done yet. Where exactly are you? |

00:23:28 | Right now? |

00:23:29 | At which problem are you now? Which problem did you complete? |

00:23:33 | At 27. |

00:23:35 | And this one you haven't done? |

00:23:36 | Right, I skipped that one. |

00:23:37 | That you cannot do. You can't just skip problems and then proceed with other problems. |

00:23:42 | That is impossible, because then you don't understand those yet. |

00:23:45 | Yes, a little bit, I do find it to be quite difficult. |

00:23:47 | Yes, I can imagine. But at what problem did you end? |

00:23:49 | At 12. |

00:23:50 | At 12, so then continue with 13. |

00:23:51 | And don't do the additional exercises for tomorrow, instead you just go as far as time allows you to. |

00:23:57 | Yes. |

00:23:58 | Preferably more than two pages, otherwise you will fall behind even more. |

00:24:01 | Yes, okay. |

00:24:02 | Start with this problem and then come and see me any time there's something you can't figure out, so that I can help you move on. |

00:24:14 | I will make an appointment with you later because- yes, no, I think that you are working badly- |

00:24:17 | I already made that one- |

00:24:18 | Well, at least sit straight and get to work, please! |

00:24:21 | Teacher, I have been checking the wrong chapter, I thought- |

00:24:25 | Oh, that is smart. |

00:24:26 | :00] |

00:24:44 | May I briefly interrupt? I want the both of you to write down a clarification for yourselves at problem 27. |

00:24:50 | Okay. |

00:24:51 | So that you explain to yourself why that line is there, otherwise you will never find it back. |

00:24:57 | Okay. |

00:24:58 | Hm! |

00:25:09 | Yes, I know because I came to- A has the starting number four. And- |

00:25:16 | Shall I demonstrate it on squared paper? That way you can see it very clearly. |

00:25:20 | Yes, because that one's lower, that other one. |

00:25:23 | Look, have a look. I will put it on top of the squared paper. I have put squared paper underneath it. |

00:25:32 | Are you able to see now... |

00:25:40 | Can you now figure out for each line what the starting number is, what the vector- the slope is and what the equation is? |

00:25:50 | Why don't you close problem 27 in your book and grab your notebook and write down the equations of these four lines. |

00:26:01 | Equations? |

00:26:02 | Uh huh. |

00:26:05 | Okay, um- |

00:26:07 | Y equals... |

00:26:11 | Simon, I also have a nice little assignment for you. |

00:26:13 | Y equals four- |

00:26:15 | Oh yes. |

00:26:16 | But I am busy correcting. |

00:26:18 | But you had problem- but you didn't understand problem, uh, what was it again, 23? You weren't able to figure that one out either. Huh? |

00:26:24 | (inaudible) |

00:26:25 | You didn't have that one, right? |

00:26:27 | Here I have drawn- Simon, four lines here. |

00:26:31 | Right. |

00:26:34 | Would you, without having your book open, be able to write down an equation of those four lines? |

00:26:47 | Would you? Would you like to give that a try? |

00:26:50 | Sure. |

00:26:53 | Yes, maybe your neighbor would like to join as well. |

00:26:54 | Yes, he does. |

00:26:58 | Are you able to do this? |

00:27:05 | Let's have a look. |

00:27:06 | I will indicate clearly where the reddest points are. |

00:27:09 | Let's see. The green one is not very legible. |

00:27:12 | :00] |

00:27:19 | Times three. |

00:27:21 | You mean: times minus three. |

00:27:23 | No, it's minus three times- |

00:27:26 | Hello! Four minus three... |

00:27:30 | No, I know that. |

00:27:33 | In case more people are having difficulties in making the equations, you could close your book for a moment and try to do this one. |

00:27:40 | You don't have to, but you may. It is good practice. |

00:27:44 | We don't really understand it. I think. |

00:27:46 | Well- |

00:27:47 | I understood it, but now I don't understand it anymore. |

00:27:48 | Well, you will not understand it until you can truly solve it. When you can really do it. |

00:27:56 | Take a line... |

00:27:58 | Okay, A. A, line one. |

00:27:59 | Line one. What do you need to know before you can make an equation? |

00:28:03 | The starting number and (inaudible) number. |

00:28:05 | Go ahead and see if you can find the starting number and the slope. |

00:28:08 | I have four and minus three. |

00:28:10 | But how can you place that in the equation? |

00:28:11 | Yes. |

00:28:12 | How do you start your equation? |

00:28:13 | Y. |

00:28:14 | Well, why don't you write that down. |

00:28:15 | That's what we have! |

00:28:16 | Well great, then let's see what you have? |

00:28:17 | I have Y is four. |

00:28:19 | Y is four and this I can't read- |

00:28:22 | Yes. |

00:28:23 | It shows a hyphen. |

00:28:24 | Yes, I had X, times X or so. |

00:28:27 | Just four X. Minus three is? |

00:28:29 | Yup, I also have that. Minus three is- |

00:28:31 | What comes after that? |

00:28:32 | Is X. |

00:28:33 | No, but that is not possible. |

00:28:34 | No, because you started with "Y equals"- |

00:28:37 | Yes. |

00:28:38 | So you've already got "equals". |

00:28:39 | Oh, okay |

00:28:40 | Y equals four X minus- minus three- |

00:28:42 | Minus three. I have that, too. |

00:28:44 | Okay. |

00:28:45 | And then what? |

00:28:46 | Well, very good. Then do the red line. |

00:28:47 | Oh, is this one finished already? |

00:28:49 | S?:00] |

00:28:52 | That is all there is to it! Because now you have the whole equation. |

00:28:55 | Listen, you can check it by saying "I have the letter X, I have the letter Y, and I have a starting number as well as a slope". |

00:29:02 | And then you are done. |

00:29:03 | Okay. |

00:29:04 | Okay, uh... |

00:29:06 | Y equals. Y equals five. |

00:29:08 | Isn't it three? We're going from line two- |

00:29:13 | Arnold? Come on. |

00:29:17 | Y equals five. Um... |

00:29:24 | Times- |

00:29:27 | Five times X minus- minus three. |

00:29:31 | No, that is not possible because, since then you've got (inaudible). |

00:29:34 | No, minus one, two, yes though, minus three. |

00:29:38 | And then? Teacher, we got the second one too. |

00:29:44 | Super. What- and what is it? |

00:29:46 | Well, I am not saying anything- |

00:29:47 | I got it, though. |

00:29:48 | Yes? |

00:29:49 | Y equals five times minus three- X minus three. |

00:29:52 | Five X minus three. Yes, very well done. And now, what's the starting number? |

00:29:57 | Five. |

00:29:58 | Okay, what is the vector number- the slope? |

00:30:01 | Minus three. |

00:30:02 | Okay, you should always begin with? |

00:30:05 | Uh, Y equals five. |

00:30:09 | Yes, but why five? |

00:30:11 | Well, because that is the starting number. |

00:30:12 | We always put that at the end. You've also put it at the end over here. |

00:30:15 | No, because four is the starting number. |

00:30:17 | Ooh. Yes, it is the wrong way around. |

00:30:21 | But you just said that it was correct. |

00:30:23 | Yes, that was silly of me, I didn't see that. No, I was looking upside down and therefore didn't see it well. |

00:30:29 | No, the- the- the starting number is always the independent variable that is written at the end. |

00:30:33 | This is what I drew, how I- |

00:30:34 | Yes, that four is added in every step. |

00:30:37 | So then this is also, this is also the wrong way around. That la- two. |

00:30:40 | Uh, four X minus three. Yes, it is the wrong way around. |

00:30:45 | Oh, and the last one too? |

00:30:46 | And the last one- |

00:30:48 | Y equals minus three. |

00:30:49 | Yes. Right. |

00:30:50 | Uh, times- |

00:30:51 | Times five. No, no, no: times X minus five. |

00:30:57 | Why don't you begin by writing down "Y equals". |

00:30:59 | Y equals minus three. Doesn't it? |

00:31:03 | Yes. |

00:31:04 | X- or no, minus four, four, X four. |

00:31:12 | It is a positive four. It is on the upper side and a positive four is a plus four. |

00:31:18 | If the starting number is negative then it is a negative number and then it is minus four or minus three or whatever... |

00:31:26 | But when it's on the upper side than it is plus. |

00:31:28 | Yes, but you don't have to write the plus down? |

00:31:30 | You do write it down because it has to be added on to it. So what do you get? Write it down again, write it down again. |

00:31:36 | This number one? |

00:31:37 | Yes? |

00:31:39 | Y is- |

00:31:40 | Yes? |

00:31:41 | Minus three- |

00:31:42 | Yes? |

00:31:43 | X, um, minus four? |

00:31:48 | It is located on the upper side- |

00:31:49 | Oh no, it is four, four! |

00:31:50 | And, something has to be inserted. |

00:31:52 | Yes, but that's what I can't figure out. |

00:31:54 | Is it a positive four or a negative four? |

00:31:56 | Oh, plus four. |

00:31:57 | X plus four. |

00:31:58 | Yes! |

00:31:59 | But in the book it is shown differently. |

00:32:00 | No. Just write it down this way for now. This way is correct. |

00:32:04 | Three X, and I put this here. |

00:32:06 | Correct! |

00:32:07 | Okay. |

00:32:08 | All right. Now the next one, now the red one. |

00:32:11 | Is it working or not? |

00:32:14 | Yes, with the red one. |

00:32:15 | You got the red one? Tell me about it. |

00:32:17 | That is the (inaudible). |

00:32:21 | No. |

00:32:24 | No, because that's the slope! |

00:32:27 | Look, you have the slope correct and the starting number correct except the starting number is always written at the end. |

00:32:36 | This is how that's written. |

00:32:38 | Y equals, and then over here you put the slope, and that you multiply with X. And then over here is the- |

00:32:52 | Starting number. |

00:32:53 | Starting number. |

00:32:58 | Yes? And you have- this is the slope. That goes over here- |

00:33:02 | That has to be the other way around. |

00:33:03 | Right. Well, yes, but that is as different as day and night. It is a totally different equation. |

00:33:06 | Yeah. |

00:33:09 | Why don't you write it down correctly. |

00:33:11 | (inaudible) |

00:33:12 | You better start over again otherwise it- it'll get so messy that you will never be able to find it back. |

00:33:28 | Shh. Keep going, I will be right over. I'll be right over. |

00:33:36 | Yes, and over here you still need to add a- a symbol in between here. Is it a positive five or a negative five? |

00:33:43 | Uh, negative. |

00:33:44 | Let's see, where is the diagram? |

00:33:46 | Right there. |

00:33:47 | This red line. Here is the starting number. Is it a positive or a negative number? |

00:33:52 | A negative number because it goes down. |

00:33:54 | I am talking only about this point. |

00:33:57 | Positive. |

00:33:58 | It is positive because it lays above the axis. If this were the intersection then it would have been negative. |

00:34:01 | This is positive. So it is a positive five. What you have is a negative five. Now go ahead and make the- you want to have a plus here. |

00:34:13 | Don't just simply remove it because then there's nothing there anymore. Plus five, it needs to be added on. |

00:34:18 | Now do the black one. |

00:34:20 | Teacher, you are not explaining it correctly because the book says something else. |

00:34:25 | Yes, well, the book has a different sequence but it is correct, though. Look what- which one did you have? This one. |

00:34:31 | Y equals minus three X, that's what you guys have, plus four. |

00:34:36 | Except they put the four in the front because in the book they have shown it in a different sequence. The sequence doesn't matter. |

00:34:42 | Yours is correct. And I think it is much more- Arnela, listen for a second, please. |

00:34:47 | I think it would be much easier for you guys if you always used the same sequence. |

00:34:51 | What the book is trying to do is make it more complicated by changing the sequence as well. |

00:34:55 | But if you keep using the same sequence at all times, then you make thing a lot easier for yourself. |

00:35:01 | This is exactly the same thing as is shown over there, except in a different sequence. |

00:35:05 | Okay. |

00:35:06 | Yes? |

00:35:07 | So we should do it in this manner. |

00:35:08 | Yes, that is what I would do, yes. That gives you some grip on it. |

00:35:11 | Begin with the starting number? |

00:35:12 | What? |

00:35:14 | Do you have to start with the slope- |

00:35:15 | You always start with the slope. And then- write it down at least. Write it with red or some color. |

00:35:22 | Yes, there it is already! Okay, yes. |

00:35:26 | So we'll put this one down- that is also allowed, we are going to put that at the end, but look: the slope multiplied by A. |

00:35:31 | Yes, and then what? |

00:35:32 | The slope is separate. If you find it easier you may also begin with the starting number, |

00:35:38 | but you did: starting number times X. |

00:35:42 | But- |

00:35:43 | Oh, yes, that's at the end. |

00:35:44 | Correct. So generally the sequence is irrelevant. However please choose a sequence for yourself. |

00:35:51 | If you choose a different sequence every time, you will get confused. |

00:35:54 | Write down for yourself how you are now going to do it in the additional exercises. |

00:35:59 | What sequence will you take each time? |

00:36:02 | And it doesn't matter to me, you can choose it yourself. But you do have to choose one. |

00:36:06 | And maybe it would be smart to choose one that is the same for both of you. |

00:36:09 | Then you can help each other out every time. |

00:36:11 | Or spy. |

00:36:12 | For example, yes. |

00:36:15 | The slope... |

00:36:17 | And then the starting number. |

00:36:19 | Eh, two, four, six, seven- |

00:36:32 | And then? |

00:36:34 | What do you mean: And then? |

00:36:36 | Starting number! |

00:36:37 | Yes. But didn't you always have to (inaudible). |

00:36:40 | Yes? |

00:36:42 | I don't quite get four. |

00:36:49 | Whether this is linear? We have looked at two ways to find that out. The first way was? |

00:36:55 | A table. |

00:36:56 | Yes. You may do that. You could say I am going to make a table for each of them and then I'll check- |

00:37:03 | How can you tell by looking at the table whether they are linear? |

00:37:06 | When they're all equal steps. |

00:37:08 | Correct, you may do that. |

00:37:09 | And the other way we observed yesterday, well, we said to each other that we could also check which ones are not linear. |

00:37:16 | No, that is two. |

00:37:17 | Do you remember that series we just wrote down? |

00:37:20 | You have to square it. |

00:37:22 | Yes, squared. You need to memorize that. You may also do that. For instance you could say, well, B is not linear because it contains a number that's squared. |

00:37:32 | You are also allowed to write that down. But if you say you find those tables easier because that way you don't have to memorize them, that is just as well. |

00:37:36 | With that- with that letter that's not minus either. is it? |

00:37:38 | Very true, that is not linear either, very good. |

00:37:40 | C is not linear either, then. |

00:37:41 | C is not linear either, no. |

00:37:43 | And the other two, if you- if you have a good look, you can recognize it because you may see something that looks like a starting number and a slope. |

00:37:51 | Hers is not correct, of course. |

00:37:52 | Then you already have an idea as to which ones they are. But then you still need to explain it. |

00:37:57 | Either you use this method where it shows a number squared which tells you it's not, |

00:38:01 | or you do it with a table and you say that each time equal amounts are added, so therefore it is. |

00:38:04 | Okay. |

00:38:05 | Yes? |

00:38:06 | Y is: minus two-and-a-half X- minus two-and-a-half X and the starting number is? |

00:38:24 | Do you have the black one already? Yes? |

00:38:31 | Well, let's see. Y equals- so far so good. |

00:38:35 | The starting number is four. |

00:38:36 | Let's see about that. From that- what line did we have here? |

00:38:39 | The black one. |

00:38:40 | The black one. The starting number is four. That is correct. Very good, and now the slope. |

00:38:46 | I thought it was minus one-third. |

00:38:47 | That's kind of hard to see because here because it doesn't have any squares underneath. |

00:38:51 | But over there you can see it, right. If you go one step to the right then you have to go three steps down to get back on the line. |

00:38:58 | Well, down below it is negative- they run parallel to one another so they must have the same slope, |

00:39:03 | the same direction, so their slope is the same. |

00:39:06 | So that is also negative three, yes? Okay. One is kind of difficult so I would skip that one for now if I were you. |

00:39:19 | Um, I think you are able to do the blue one. |

00:39:28 | Shall I, um, with the blue one I will- let's see, is that right? Or do we have more here? Um, |

00:39:40 | Well, this one is also not that easy after all. |

00:39:41 | Just go ahead with the additional exercises because you will come across them here again. Yes? |

00:39:46 | All right. |

00:39:50 | Draw it with pencil and a triangular protractor. |

00:39:53 | Teacher, do I have to- so far I have as starting number (inaudible). |

00:39:57 | No. How do you always find the starting number? |

00:40:01 | Um, where it, where it intersects. |

00:40:03 | What line does it have to cut, and it's not intersecting, it is cutting? |

00:40:06 | Oh, the, the... |

00:40:09 | Yes, yes! |

00:40:10 | One, the one. |

00:40:11 | One. Very good, yes. |

00:40:13 | And then the slope is two. |

00:40:17 | If you go one step to the right- |

00:40:20 | Yes well, aren't you, aren't you supposed to go down like that? |

00:40:22 | To the right, always one step to the right, what are you supposed to do to get back onto the line? |

00:40:31 | So over here I am going to go one to the right. What should I do to end up on to the black line again? |

00:40:37 | One half, for example? |

00:40:38 | One half, very good. So upwards is always positive, yes? |

00:40:45 | Madam, (inaudible). |

00:40:46 | Really? The other one, the, uh, the blue one and the green one are quite tricky and I wouldn't give those on a test, with tricky number like that. |

00:40:54 | I will have a look at those tomorrow, yes? |

00:40:56 | All right, all right. |

00:40:57 | See you all tomorrow! |

00:40:58 | See you tomorrow. |

00:41:00 | I would still like to collect all of the answer booklets! I only have one, I have two, I have three, I have four, |

00:41:10 | and I am still missing two! And I would still like one! Number two still missing! |

00:41:22 | Who still has one answer booklet? |

00:41:36 | Yes, very nice, thanks. See you tomorrow... |

00:41:40 | (inaudible) |

00:41:41 | You have to see how far you'll get. |