[speaker001:] to to do
[John:] To build up a dictionary to see how words are used. Because you see erm say if you look at business letters, erm people say I should be very grateful if. People don't talk like that, they say I'd be grateful. they don't talk so the language changes all the time, and each time they bring out a new d dictionary they try to say erm this is the way people talk, and bring it up to date so that we're not all talking in the past. And new words as they come in and new meanings of words like people might say erm, Oh I'm I'm well pleased. or something or they'll say you know is it any good they'll say, Oh it's really wicked. And you think, Oh what does this mean? and you look it up in a dictionary wicked and you think, Oh
[Ian:] [LAUGHTER] Yeah.
[John:] that doesn't sound too good. But there's a so there's another meaning comes into it and they update the dictionary like this and they they listen to the tapes and find out how people use it, and how language changes. That's all it's about.... Now what were we looking at last time?...
[Ian:] Big nasty
[John:] Oh that exam that I thought was probably
[Ian:] Yeah.
[John:] erm
[Ian:] I done it.
[John:] a bit difficult.
[Ian:] I done it, I'm just trying to find it.
[John:] And you did very well on the geometry didn't you?
[Ian:] Yeah. I I got to that erm question you know you said couldn't do any the geometry do the rest.
[John:] Yeah.
[Ian:] So I got up to that one but...
[John:] You've been doing Venn diagrams as well?
[Ian:] Yeah.
[John:] Is that is that recent or is that the old stuff the Venn diagrams?
[Ian:] What do you mean by recent?
[John:] Have you done it in the last sort of few weeks?
[Ian:] Oh yeah it's the last few weeks.
[John:] Good. No problem with that?
[Ian:] No.
[John:] No?
[Ian:] Been learning that in school.
[John:] We we did one didn't we a lot long time ago did we do one on football? Erm who's supporting Liverpool and Everton.
[Ian:] No we didn't never gone over the Venn diagrams.
[John:] Okay. Well if if you're happy with Venn diagrams fine leave it. Erm... I suppose got here a bit later you would have had more time to sort this lot out.
[Ian:] . I can't find it anywhere.
[John:] Okay.
[Ian:] I I did do it
[John:] It's a I mean no it's up to you if you say you've done it that's fine by me. Erm do you think you got it all sorted out, no problem with it?
[Ian:] Yeah.
[John:] Great.
[Ian:] Apart from one question I think it was question nine.
[John:] Right let's
[Ian:] But
[John:] have a look at that
[Ian:] I'll try and get the paper. Hang on.
[John:] Oh that was that was it wasn't it because they had erm the pentagons joined up. [whispering] Is that your answer to it? [] No it was different one that's
[Ian:] No that one that's a different
[John:] that's a project is it?
[Ian:] No this this is this is just a maths paper. We've got a big erm like cupboard in in the school and they've they've got loads of paper threes paper twos. So I
[John:] So you
[Ian:] helped myself to a load of paper threes and I seem to have done them all. I wiped the school clean of paper threes. [LAUGHTER]
[John:] [LAUGHTER] That's great. So whatever sort of question comes up you will have seen something like it wouldn't you?
[Ian:] Yeah.
[John:] Well that'really great.
[Ian:] Ah is this it? That's it
[John:] That's it.
[Ian:] and this is
[John:] your answer.
[Ian:] my answers. I think.
[John:] Ah.
[Ian:] Is it?
[John:] Okay?
[Ian:] Hang on.
[John:] Could be. I I I'll have a look at the question.
[Ian:] No that's for that one unfortunately.
[John:] Ah that be That's one two three. Right okay.... Number nine, transformations. Translations and rotations and reflections. Is that the one that you had the problem with?
[Ian:] Yeah.
[John:] Okay.
[Ian:] hang on. Hang on. Yeah that one and number ten, these parts here.
[John:] Okay. Using a graph. Er where should we start?... Start with number ten I think. Because graphs particularly distance and time graphs I mean both of these are very likely to come up, you'll get a translation of some sort, and you'll get a distance time graph but the the the graphs or... distance against time er are useful in other problems as well. Erm yeah we'll look at that.... Okay don't worry about the ones you did, if you're if you're happy with them
[Ian:] Yeah.
[John:] erm it might be an id if you can find them for next time
[Ian:] Yeah I will yeah.
[John:] So I can just have a quick check and make sure they are okay, but I'm pretty sure if you think they're right
[Ian:] Yeah.
[John:] they'll be right. So... What did you make of this number ten then?
[Ian:] I think first part to it, but... then er... it seemed to go downhill so to speak....
[John:] So you had a formula... you had a formula for calculating... the value.... I haven't read the question I saw V equals and I thought it was velocity. It's the value of a car. Number of miles it's travelled.... Okay erm can you tell me in words what that formula means really?... Erm if the car has done a lot of mileage, will you get a lot of money for it or will you get less money?
[Ian:] You get more money.
[John:] let's say the car's done no mileage at all, how much would you get for it?
[Ian:] Nothing.
[John:] You get that. You'll get nine thousand five hundred okay?... If it's done a thousand miles, you'd get nine thousand five hundred and then cos of the mileage they'd take off a thousand divided by ten, so they'd take a hundred pounds off that. You only get nine thousand four hundred.... Okay? Are you happy with that?
[Ian:] Yeah.
[John:] Right so... I won't draw a graph, but we'll see if we can work out a sketch of a graph, roughly what it's like. Erm what's the value going to be doing? This is this is the mileage going along there, that's the number of miles that the car has done, and that's its value. As the miles get more and more and more is the value going to get bigger or get smaller?
[Ian:] Smaller.
[John:] Right so it's going to be something like I mean this isn't anything like to scale but it's going to be something like this perhaps.... Okay. It starts off at some value here and it goes down until eventually when it's done a certain number of miles the car's worth nothing. Yeah? So if we can find out where those these are the important points, where it cuts that axis and where it cuts that one. So... this point is where it's done no miles. How much is it worth when it's done no miles, well they give you that one anyway. It'll be nine thousand five hundred, take off N times zero over a hundred. Well don't take anything off. So it's still worth the nine thousand five hundred. got nine thousand five hundred up there. Now... complete so what do they want us to do? Complete that table, plot a straight line graph and use your graph to find these values. Okay.... We've got V is equal to nine thousand five hundred, that's how much it's worth, and then take away... N over ten. Take away... a tenth of a pound for every mile. Yeah? So when will it be worth nothing? When will it have gone right down to nothing?... In other words when is when is V equal to zero? Oh we've got a an equation for V. So wherever we've got V put zero. Zero equals... nine thousand five hundred minus one tenth of N. Are you happy with that? This this expression here V nine thousand five hundred minus a tenth of N that always tells us what V is worth, anywhere along here it'll give us the value of the car. Now what if... what happens as N gets bigger and bigger as it's done more and more
[Ian:] The value
[John:] miles.
[Ian:] The value goes lower.
[John:] Right every time N gets bigger it means we're taking more and more of this nine thousand five hundred. So if it's done ten thousand miles, we'd take a tenth of ten thousand, we'd take a thousand pounds off this. If it had done nine thousand five hundred miles, how much would we take off?... hey?
[Ian:] Ninety five.
[John:] Nine nine nine fifty yeah Okay nine hundred and fifty divided by ten. Oh that would still leave us with something left over. So the value is getting less and less and less. The value is always equal to, at any point it's always equal to nine thousand five hundred take away a tenth of the number of miles it's done. So when does the value come down to zero, well it comes down to zero when the bit we're taking away... is equal to that. Yeah? So we'll just use this equation. Just say V is equal to this but we want to find out when nought is equal to it. Nine five O minus would it help if we used X instead of N?
[Ian:] No
[John:] No? Okay. Minus one tenth of N, so you've got an equation there and we want to find N. But we've got a fraction in it,... so what would you what would you do? What would be the first what would be the first thing you'd want to do?
[Ian:] Get rid of
[John:] The fraction.
[Ian:] the fraction.
[John:] How could you do that? You have a go and see what what would you multiply by to get rid of the facti fraction.... Or maybe maybe that isn't the first thing you want to do, if you want to do it a different way, do it whichever makes more sense to you....
[Ian:] I wouldn't know how to go about starting, you know trying to get rid of this fraction.
[John:] Okay if we got if let's let's leave the fraction there for a moment then. Just write the equation out again... and let's get the we'll write it as N over ten shall we? Let's get the N over ten onto this side. So what would we need to do to make that disappear?
[Ian:] Add it to both sides.
[John:] Add add N over ten to that side and add N over ten to that side, okay if you want to tidy it up and see what it comes to. What's that going to come to?
[Ian:] N ten.
[John:] N over ten on the left yeah....
[Ian:] and ninety five.
[John:] And nine thousand five hundred. Ninety five hundred yeah. Okay. Now how could you get rid of
[Ian:] Mm multiply ten by both sides.
[John:] Exactly multiply both sides by ten.... So how do you multiply by ten? The what's the quick trick for multiplying by ten?
[Ian:] Add another nought
[John:] That's it just add another nought on.
[Ian:] Ninety five thou
[John:] So it's ninety five thousand. So when the car's done ninety five thousand it's worth nothing. So we've found the other bit on our graph now. We know that up there it goes to nine thousand five hundred and along here... it gets to ninety five thousand. Now... where it goes negative a negative value I don't think we'd be interested in that, that means that the car's got to the point where you have to pay someone to take it away. And the more miles it's done, the more you pay them to [LAUGHTER]. So it doesn't make a lot of sense after that point the graph. So now we know... what's going on here what the what the graph's going to look like. We've got some others to fill in. It tells us this this isn't doing exactly what it said here right. What it says here is... zero... N is zero, the value is nine thousand five hundred. When N is two thousand, what's the value? Four thousand, six thousand, eight thousand okay. And you plot the graph. Erm we'll use those points as well. Now this techniques we've been using here is for you can use that for any straight line graph. What happens when the one that's going along there the the X normally in this case the number of miles. What happens when that's zero? Well that's the value. And then the other one is well what value of X would make Y zero? What number of miles would make the value zero? Then we'll get those two points and that'll give us our straight line. And when we calculate these other points they should fit on it as well. Are you happy with that?
[Ian:] Erm. Yeah I understand it li like when you're teaching me but I don't think I I I understood you know how you were doing it.
[John:] Okay. That's that's that problem. Let's do a different one.
[Ian:] It's like the graphs you know
[John:] Right.
[Ian:] I'm not too sure on graphs. You know like
[John:] Okay. This this is
[Ian:] when when they stop and the time goes on what's the time between this that they go
[John:] Right.
[Ian:] What's the speed he's travelling at?
[John:] Yeah this is the This comes in quite a lot. Spee distance time graphs and speed so let's let's have a look at er
[Ian:] Do you want to use
[John:] a different I'll just I'll just sketch them on here then you can when we get one sorted out you can draw one on there. Let's have a look. So here's a graph and we'll say the distance... let's see the distance is equal to erm a constant speed let's say he's doing sixty miles an hour. Well we won't write S we'll write distance. The distance he's travelled is equal to sixty miles per hour... times the number of hours, he's been travelling for okay? So he's driving along the motorway at steady sixty. Erm how far would he go in one hour?... Steady sixty miles
[Ian:] S sixty miles.
[John:] How far would he go in erm half an hour?
[Ian:] Thirty miles.
[John:] And in two hours?
[Ian:] A hundred and twenty miles.
[John:] Okay. So we want to draw a picture, that's all a graph is, a picture so it's eas so we can... get a good idea of what's happening and we can also read off at any time. So we'll have up here, how many miles he's going. Erm... fancy breaking the speed limit. So shall we change it make it a hundred miles an hour. Okay he's doing a hundred miles an hour. Along along the bottom time if time is one of the things you're working with it just about always goes along the bottom.
[Ian:] Yeah.
[John:] So this is this is the time... in hours. Up here is how far he's travelled in hundreds of miles say. One hundred, two hundred okay.... And five hundred, six hundred. Now what happens when he's just starting off from home? Well the distance is equal to a hundred miles an hour, times how many hours has been travelling for? None. we've only just started off. So when time is at none, zero, we haven't started yet, how far is he he's zero, he hasn't moved anywhere.
[Ian:] Mm.
[John:] So there's one point. How far has he got after two mile after two hours? Distance equals one hundred times the time. So
[Ian:] two two hundred miles.
[John:] He's gone two hundred. So after two he's gone two hundred. Erm how far's he gone after six hu after six hours?
[Ian:] Six hundred miles.
[John:] Right so after six, three, four, five, six, he's gone six hundred. Okay now if we join those up,... I've got one here it's okay. If we join, we've got three points now once we know it's a straight line, three points is enough. Two is enough really but just to be on the safe side use three and if they don't all lie in the same straight line then one of them's wrong. You don't know which one you'd have to check all of them again. So we've drawn the graph and now we don't have to keep calcul I mean this is easy it'probably easier to work it out in your head then to especially with these figures. But say the figure had been erm fo he's travelling at forty eight point three miles an hour, and you want to know how far he's gone after twenty seven minutes or something, it's a bit more awkward then so it'd be easier to look it up on a graph. Once we drew this picture. Up there is how far away from home he is in miles along there is the time he's been travelling for. that's travelling time. So without working it out, but using the graph now, how far would he be after three and a half hours? Three and a half is about there. Look up there, what do we get? Actually it comes out to four hundred cos I haven't drawn these properly come up come out to three fifty. Let's draw those in properly. One two and so on like that. Erm that was an easy one that was a nice simple one. Well now I'd like you to do one.
[Ian:] Er.
[John:] Do the hard I do the easy one you do the hard one. This time it's the distance he's travelled is given by erm... What speed shall we let him go at? Twenty five miles an hour say. Twenty... let's make it twenty miles an hour. This time he's a cyclist he's a racing cyclist and he's keeping up a steady twenty. So it's twenty miles an hour times the number of... So the distance which is in miles is equal to twenty miles an hour times hours.... Okay could you draw a graph of that?... And use you're you're not going it's not going to be negative is it?
[Ian:] No.
[John:] So use all your all your paper go down quite a bit. Just leave yourself a little bit to write underneath it.... You're going to need some sort of scale well.... Miles per hour.... So how many squares have we got up here? Roughly. [whispering] twenty or so. []
[Ian:] Mm thirty.
[John:] [whispering] About thirty okay. [] Let's say let's make it erm one square going up there is ten miles. Okay so you could mark that off in ten in ten mile steps.... And we want to know let's say we're interested in a a time of about ten hours or something.... Okay that's fine. So... along there if we have two squares is equal to one hour.... Now... we've got a formula, so we build up a little table. there okay something like this. A good one to start with is so if we put time along the top here. Yeah? Erm T and underneath we'll have distance. A good one to start with is what happens when the time is equal to zero? If he's been travelling for for zero hours, how far has he gone?
[Ian:] No miles.
[John:] No miles okay. Erm if he's another easy one is if he's been travelling for one hour how far has he gone?
[Ian:] Twenty miles.
[John:] Twenty miles okay. And if he's been travelling for ten hours, how far's he gone?
[Ian:] Two hundred miles.
[John:] Right okay. Now we've got three points there it's probably they they give you four here so we'll do another one erm, when he's been travelling for five hours how far's he gone?... So it will be for that one it'll be
[Ian:] One hundred.
[John:] That's it yeah. Great so it's twenty miles an hour times five hours equals one hundred miles. So now we've got a table and you've got four values in, which is plenty. So if you'd like to plot those.... Okay now check they all lie on a straight line. If they don't then one of them's wrong. They do.
[Ian:] Yeah.
[John:] That's a very good... indication that you got everything correct. It might not be cos you might have got the formula wrong and you've got every one of them wrong but it's usually a good sign. So just join those up.... And now you can ask I can sort of ask you questions on it like if he's been travelling for let's say If he's if he's covered sixty miles how long has he been riding for?... So you think you can use that set square actually which helps.... He's he's travelled sixty miles... come down that way...
[Ian:] Three hours.
[John:] He's been going for three hours. Okay erm If he's done two hundred miles, how long has he been riding for?
[Ian:] Ten hours.
[John:] Okay now looking at it the other way, you see we we had those points but we don't know what happens over here. If he'd been cycling for fourteen hours, let's say fifteen hours.... If he'd been cycling for fifteen hours, you you have a look and tell me how far he would have gone....
[Ian:] Three hundred miles.
[John:] Okay now any problems with that?
[Ian:] No.
[John:] I think you understand that
[Ian:] Yeah.
[John:] very well don't you, that's all there is to it. It's use your equation to make a little table. What happens nought is a good one to go for. Now what happens when he hasn't done any time at all and time is just starting off. Where is he? Well usually usually not always but just about always he's at home he hasn't gone anywhere so how far away is he zero as well so you get zero zero goes through the origin. And then you just put the numbers in for different hours into the equation into the formula that tells you how far he's gone, work it out. So let's try another one and I'll give you some of the details and the questions that I'm going to ask on it later. Now with this one the one we've just done you don't really know how far to go I mean I might have been going to ask erm how long has he been travelling if he's been riding for a thousand hours. And you're your graph paper [LAUGHTER] [] few metres long, or you'd have to cramp this scale up quite a bit bring it closer together. So we'll do something similar if erm the train... Shall I make it very awkward?
[speaker001:] Yes.
[John:] Yes. The train leaves... this is an express it leaves Euston at some time, we don't know when, at some time. Okay and it doesn't stop anywhere it's going all the way up to Scotland before it stops. And it does a steady what shall we have? Sixty miles an hour, eighty. Eighty miles per hour. Right. Hello I'm sorry to mess up your tea and everything. I was explaining I got
[speaker001:] Oh no it's it's alright.
[John:] so many changes in lessons and everything today
[speaker001:] Yeah.
[John:] had something else that I was just doing and I thought Oh I should be at Ian's for five o'clock. I didn't look at my timetable which I've got with me.
[speaker001:] Yeah.
[John:] [LAUGHTER] And I got here early.
[speaker001:] It's alright he'd practically finished anyway it was alright.
[John:] Hello hello hello.
[speaker001:] Oh yes nosey. Come on.
[John:] late and I was early.
[speaker001:] [LAUGHTER]
[John:] You alright. Mm? You're a cute little thing aren't you?
[speaker001:] Come on Taff. Out you come.
[John:] Cheerio Taffy.
[speaker001:] Out.
[John:] See you later.
[speaker001:] Out.
[John:] that way
[speaker001:] Out.
[John:] Go on shoo. [LAUGHTER]
[Ian:] Thanks for the coffee thanks.
[speaker001:] Okay.
[John:] It goes a steady eighty miles an hour right. Now when it's been travelling for three hours... it goes through Manchester.... Okay... and when it's been travelling for five hours... it goes through erm what's another station up there? let's say Carlisle something like that. And when it's been travelling for seven hours it gets into... Glasgow.... Okay. It goes it goes through Manchester and it goes through Carlisle it doesn't stop. Doesn't stop from Euston it doesn't stop till it gets into Glasgow. Now what we want to do is try and draw some sort of picture for that. Any ideas?... What's happening there? What we're going to try and find out erm so instead of having these three hours and things we'll say at at three o'clock... it goes through Manchester... and at five o'clock... it goes through Carlisle. At seven o'clock it goes through it gets into Glasgow. Right now we want to find out what time it les it left Euston. And we want to know how far is it between Carlisle and Glasgow, between Manchester and Glasgow and between Euston and Glasgow. Pretty impossible task eh? And we're going to do it with a graph. Any ideas what what could you look at that sort of question and you think this is impossible. Yeah? No chance of working any of this lot out. But they say draw a graph. That's all they's told you. A steady eighty miles an hour and where it was at these times. So... We know the graph is going to look something like this.... There's time... and there's distance.... And time seven o'clock okay? [whispering] [] Seven o'clock when it gets into Glasgow and five o'clock when it gets into Carlisle and three o'clock when it goes through three o'clock when it goes through Manchester.... Mm we don't know what what what could we put in? Could we put part of the graph in do you think?... How far apart are Manchester and Carlisle going to be?...
[Ian:] Hundred and sixty miles.
[John:] Right it takes two hours it's going at a steady sixty, great, so we know that's a hundred and sixty. And what about the other two Carlisle and Glasgow?
[Ian:] A hundred and sixty.
[John:] That's a hundred and sixty okay. Erm and we need some more information there but let's say we know that Euston to Manchester is a hundred and sixty as well okay? Let's say that's erm Euston to Manchester is a hundred and sixty miles, so where would Euston be on this?...
[Ian:] It'd be there.
[John:] Okay so that would be at one o'clock and there's another hundred and sixty.... So those mileages go up here. Now if if we start up here somewhere.... Let's draw the line in first.... can decide what this is. Erm... From there to... there let's make it er...... Okay that's a hundred and sixty, that's a hundred and sixty, that's a hundred and sixty... That's seven o'clock and that's five o'clock, three o'clock... and that's one o'clock. So... that's what the graphs going to look like. Now you wouldn't get one as hard as that in an exam. But I think you could do that sort erm provided you didn't sort of panic no chance of doing this okay? What what do you think of it now? That sort of question. You still think it's
[Ian:] What I've you know when you're like given a plain axis and I have to decide what to put on the sides I I'm sometimes a bit
[John:] So is it which one goes top and which goes along? Is that okay, to work that out or not sure about that?
[Ian:] I'm not sure
[John:] Okay. Well there's an easy way to divide that. If they give you an equation like this let's say the weight you're making up a a load for a lorry and you're putting erm crates on with car engines in, and the car engine weighs erm I dunno say it weighs a hundred kilograms. Erm what will the... the equation you get will be something like this, the weight is equal to... a hundred times N okay where N is the number of engines that you've got on. Okay so you could work that out for well how much would it weigh if we put five engines on? Yeah? Or how much would it weigh if we put ten engines on? Now when you see the equation like that, the one that's on the left hand side. the weight equals, the one on the left hand side that equals something that you have to calculate. The weight is the one that goes up there in the Y direction. And the one that's part of the calculation goes along the X axis okay. So this one you'd mark off... one two three... and this they would actually use this erm... if they were loading lorries they could use this sort of thing. Er so... one two three Could you dr could you draw that one then? I'll cover that up again.... Just draw a graph of that. Weight is equal to a hundred times the number of engines. The weight in kilograms is a hundred times the number of engines we've got. Erm whatever whatever units will fit the paper.... So do do it on your graph paper okay.... Turn that over.... And let's say the maximum number of engines we can ever have will be up to about twenty five.... So use use a bit more.... Take you can take it right long to the end. You don't need to go actually to the end with your numbers but you can take your line right along there cos you might find at the last minute there's an extra one you want to put on. So I'll just give you the equation you can write the equation ion the on the graph paper.... W equals a hundred times N.... Where N is the number of engines and it's the weight in kilograms. Now good that's a really brilliant point on any graph to put the units in because they give marks for that, when they see your graph they want to see you've put the units on. And along there would be... number of engines N.... Now the maximum number of engines you you're going to have we know is going to be twenty five. So how long have you got along there roughly?
[Ian:] Fifty.
[John:] About fifty. Have you got fifty, cos you had thirty last time didn't you.
[Ian:] Yeah that one.
[John:] Okay and we can get about fifty there can we? Erm... Three squares to the centimetre and we've got... twenty five yeah so we'll get erm we'll get fifty on fifty squares so twenty five engines to fit along that space of fifty....
[Ian:] One every centimetre
[John:] Mm. One every centimetre okay that'll make it one every three squares. Three squares is a bit awkward to deal with. That's that's good that's the best scale for getting it very accurate, that's great. Erm it might be better to go for two squares is one engine and they'll fit on okay. So if you mark off that... every every two squares is probably one engine.... So you've put your marks every two squares, but you don't need to write all the numbers in you could just write say erm... two, four, six, eight or you could just write five, ten, fifteen, twenty, twenty five. And that would probably be a good enough start just put five, ten. So put those in.... Oh hang on hang on hang on. Yeah these are ones okay that's one, two, three, four, five.
[Ian:] Oh I see what you mean I thought
[John:] Okay No sorry I didn't make that clear erm mark every fifth put the number on every fifth one. Five, six, seven, eight, nine okay. [tape ends] to get so you've worked out your scale for this one that goes along the X axis. What's the biggest weight you're going to get if it was twenty five of them what would it come to?
[Ian:] Twenty five hundred.
[John:] Twenty five hundred. And how far how many have you got along there roughly?... Okay sixteen times three you've got almost you can almost get fifty up there wouldn't you. If you'd have brought your scale down a bit you would
[Ian:] Yeah.
[John:] have got it on. But okay maybe we won't go to twenty five. So use the same up there. Mark every second square, and each one's a each one is a hundred kilograms.
[Ian:] Every second square? Like
[John:] Yeah.
[Ian:] that's a hundred, two hundred
[John:] Yeah.... Okay.... And the sec yeah be very careful when you're marking each second one because that one
[Ian:] Oh ooh.
[John:] Right it is very easy to do you just get into a nice little plodding along
[Ian:] Yeah.
[John:] and you're not thinking and the next minute you do that and your graph goes all out then. That's it.... So you might think why don't I make it easy and make it one square. Because they're not one of them will be two squares and one of them bill we be one and so you need to get used to working with them like that. Okay... now you need to work out a table. Ah you've got W equals a hundred times N. So we do a little thing like this.... And we put N on the top... right. N and W. So what values are you going to try? I mean they will normally give you them but erm what would be an obvious one to go for?
[Ian:] N's nought.
[John:] Right okay when N is nought when there's no engines what's the weight?
[Ian:] Nothing.
[John:] Right.... So there's a there's a good point I mean very often you'll get this point in the origin not always though so you have to watch that. And how about when erm you've got ten engines what would that weigh?
[Ian:] One thousand.
[John:] Okay. And when you've got twenty engines.
[Ian:] the squares.... Yeah?
[John:] Yeah. When you've twenty engines... what will that be?
[Ian:] Two thousand.
[John:] Okay.... If you use a ruler or some sort of straight edge it really does help because it's very
[Ian:] Yeah.
[John:] easy when you go right across from one end of the page to the other to be a square up or a square down. And then you can't work out why your graph seems to have a kink in it. Okay. Are they in a straight line?... Mm.
[Ian:] No.
[John:] So something's gone wrong. Okay. One of those points isn't right so let's have a look check them again zero... zero that's alright.
[Ian:] Yeah.
[John:] What's the next one? Ten... when it's ten when we've got ten engines the weight should be one thousand. So there's ten and if you use your set square like that go along the ten and to the one thousand. That's okay.... Right twenty... Ah not quite on the twenty.... Two thousand... yeah it is it is. Right twenty two thousand. So the points seem okay what's wrong? they're not in a straight line. They should be cos it's a straight line equation. What's going wrong? What else could be wrong?...
[Ian:] The erm size
[John:] Right so this this you made here... when you you suddenly went up one instead of going up by two, so check the scale now. Er let's check the bottom one. Let's see how that's going. So each one of these is one engine. One so maybe you can write them all on now go along and write them on. One,... Okay so that looks fine. Now what's happening up here, we're going up in hundreds. So do they go up did they all go along two squares at a time? Yeah okay. Do these all go up two squares at a time? One, two, three, four... Yeah... Now what's gone wrong here?... Ah right
[Ian:] down.
[John:] Go on have you spotted anything yet?
[Ian:] Down here it's one, two, three, four, five, six
[John:] Right.
[Ian:] and then seven?
[John:] And then suddenly
[Ian:] Nine and then it goes
[John:] suddenly
[Ian:] two hundred.
[John:] Right you started so you're going up in hundreds, it's two squares at a time, but only one hundred for each two squares. So you're quite happy with that, one hundred, two hundred, three hundred, four hundred, five hundred, six hundred oh I can do this standing on my head, right and you s you stop concentrating so much, and you've gone up to, so that should be a seven hundred
[Ian:] Seven.
[John:] okay. So if you'd like to correct the other's now.... Sixteen hundred. Right now this one this one's going to be off our scale anyway so let's pick fifteen, N is fifteen. So go through and check them again now.
[Ian:] Nought
[John:] No engines, no weight that's okay, let's put a just leave that and put a tick on it. Now the next one ten and one thousand.... Ten engines...
[Ian:] Yeah.
[John:] Yeah... okay so we can just sort of scrub that out or something.
[Ian:] Fifteen fifteen hundred.
[John:] Okay put that one in.... [LAUGHTER] Right now we'll scrub that one out, that point's rubbish now. Now try them see if it's a straight line.... Right.
[Ian:] Yeah.
[John:] Okay? So the first time you did it you were pretty happy with it weren't you?
[Ian:] Yeah.
[John:] that's okay that's got that. You'd done the hard part correctly, calculating what the weight would be for so many engines. Working out your table, your table was correct nothing wrong with that. But... because the scale was this funny thing of well two of them means one hundred, you're going along two at a time and it happens so often.
[Ian:] I don't even know why I done that.
[John:] You don't, nobody knows why, I've done it myself. Nobody knows why they do it but you do. You're going up two squares at a time and you're going one two three and it's usually when you get to around where I tend to do it is erm sort of eight and ten. I go one two three four five six seven eight ten twelve fourteen sixteen and I'll I'll think this graph's rubbish, oh I've got the scales wrong. It's so easy to do. So
[Ian:] Yeah.
[John:] check it when you've done your scale. I mean you've done it you think, Well I know it's right, there's no point checking it, I've just done it. I wouldn't do it deliberately wrong but it's so
[Ian:] It was just I I was just looking at there them down there.
[John:] Yeah okay.
[Ian:] And I was thinking, Oh what's going on?
[John:] But I wanted I mean I'm glad you made the mistake because I wanted you to do it, because I wanted you to do it because as i you know you might you might get it right all the time and then do that in the exam.
[Ian:] Mhm.
[John:] Erm much better to get it wrong now and you can see
[Ian:] Why
[John:] No one knows why you do it I mean you you know you know what you should be doing, and it's not difficult it's just so boring. It's so repetitive marking your scale and people just lose concentration and I can do this you know and the next minute you you you've gone up in twos instead of ones. Happens a lot. So you've got your graph now and then they'd have a question on it and they might say, Erm what would be the weight with five engines? It now again we're picking... Okay? Right now I'd like you to do this one on your own. It's a similar question. This time the weight... of the lorry, it's the weight of the lorry we're interested it's a say it's a small van. Erm and the weight of the lorry is equal to eight hundred plus a hundred N. Yeah? And... yeah leave it like that eight hundred plus a hundred N. It's sh shall we make shall we I'm just trying to think of making it a no I won't make it any more awkward. We'll leave it like that. Erm if you could draw a graph and use this table, N, zero, weight, whatever it comes to, erm... four... six... eight... ten.... Okay?... So if you can draw that one. The weight of the lorry, see the lorry weighs something itself so its total weight when it's got its load on is eight hundred plus a hundred times the number of engines on it. So you've got to do your table.
[Ian:] calculator.
[John:] Erm erm no. [LAUGHTER] Right you can do that can't you....
[Ian:] Yeah.
[John:] You can do the hundreds. What you can do here,... if you like... is put W there. here we could have a hundred N right and then we'd add eight hundred on to each one couldn't we. I mean you can use a calculator if you like but have a go without.... Okay?... Okay and tell me what you're thinking as you're doing it.
[Ian:] Ah what it is I want to find W so I can find that axis as well what
[John:] Okay.
[Ian:] that's gonna go have to go up to.
[John:] Good.
[Ian:] You know so I'll I'll do the table before I
[John:] Right.
[Ian:] do the graph.
[John:] Okay good.
[Ian:] Cos I might do a like a big graph like and and I only need a little tiny
[John:] and you'll little tiny bit. Or you've done a tiny bit and you need something that's not
[Ian:] Yeah.
[John:] on your graph yeah. So
[Ian:] Four times a hundred four hundred plus eight hundred is twelve hundred.
[John:] So write write it in there what a hundred N is, four hundred.... [whispering] Four hundred. [] And what about this one erm nought?
[Ian:] Nought times erm is nothing
[John:] Yeah.
[Ian:] plus eight hundred is eight hundred.
[John:] Right.
[Ian:] Yeah that's six hundred
[John:] Right.
[Ian:] ... erm eight hundred is fourteen hundred.
[John:] Right....
[Ian:] Eight hundred... That [LAUGHTER] looked like a one then [].
[John:] Yeah.
[Ian:] Erm add eight hundred... sixteen hundred thousand...
[John:] Okay and the question is now... you you could do it just by working it out with the formula. But they say, Draw the graph and then read off, what would the weight the total weight be what is W when N equals fifteen and when N equal nine. okay?...
[Ian:] Do you want me to do it by doing the graph or
[John:] Just say let's say they also want you to do I want you to do the graph er they wanted this one where it was twenty as well okay.... Twenty times a hundred what would that be?
[Ian:] Two thousand.
[John:] Right okay good....
[Ian:] There was a I ju I just looked there
[John:] Yeah.
[Ian:] I was being a bit lazy with myself then erm I just I just looked there and I was gonna double that
[John:] Mm.
[Ian:] and I I found I'd just put a hundred there instead and that's why I put two hundred
[John:] It's so easy.
[Ian:] there.
[John:] Right. It's so
[Ian:] I just
[John:] easy when you're working with figures that are going up in regular steps and they all go up in regular steps and they all go up in regular steps and the next one doesn't. And you just, Wow this is easy this, add another two hundred on or
[Ian:] Yeah.
[John:] Yeah? So watch that in an exam.
[Ian:] [LAUGHTER] []
[John:] Cos you can you could do yeah you know you could you could this question on graphs and you come out the exam thinking, Wow great full marks on that. and you say, Well what did you get. Oh I didn't get that. Oh dear er ooh. and you find that you just slipped up on the scale or even on the calculating your table because you just, Oh this is easy this is easy. and you're not really concentrating.
[Ian:] Yeah. Okay twenty eight hundred. Two thousand eight hundred right so.
[John:] Yeah.
[Ian:] W on the Y axis.
[John:] That's it.
[Ian:] So it's got to go up to two thousand eight hundred....
[John:] Awkward?
[Ian:] Yeah....
[John:] Two square to get erm... to go up in hundreds for two squares you'd want erm fifty squares wouldn't you.
[Ian:] Yeah.
[John:] Well just use one square, makes it easier.
[Ian:] One square each.
[John:] Yeah.
[Ian:] Is twenty eight squares.
[John:] It's not going to be as accurate and not going to be as easy to read it off but
[Ian:] Yeah.
[John:] if you can't fit it on your paper. Erm now is there any thing any other way you could do it? If you wanted it more accurate than that.... What you can do is you could turn the graph that way up. you still have your W going up this way and you have your N going along that way.
[Ian:] Yeah.
[John:] Okay? It might work out better that way. Could you get you say this is about fifty. Could you get fifty six along there?...
[Ian:] Probably yeah.
[John:] Okay okay so... erm lo lose that little bit there sixteen, sixteen and thirteen?
[Ian:] Twenty
[John:] Twenty nine.
[Ian:] twenty nine.
[John:] Okay.... And we're getting three squares to the centimetre so we go up there two centimetres and we get the so if we go up every two squares
[Ian:] That three squares every centimetre? It's quite
[John:] Sorry two.
[Ian:] Two.
[John:] Two you're right how did I how did I get how did I come out with three last time? Two squares every centimetre okay so we've got twenty nine so it will just fit on nicely
[Ian:] Yeah.
[John:] and we sort of get the
[Ian:] Very little
[John:] Yeah again it's not brilliant to do it this way but it will just fit on. So take that one along there. Right... and then as we go along... what have we got going along?... That goes up to about sixteen... Ooh we'll just fit twenty on. And we don't need to outside this at all cos the ones they're asking us are inside. So you'd get that one there. there's ten okay?
[Ian:] Mm.
[John:] And there's... another ten and we'll just about fit it on if we go right up to the edge of the paper sort of leave one square. And you might think well where are we going to put the scale? Well we'll put it just put it the other side of the line. It's only to how us where it is. Okay so that's nought along here going up in twos I'll let you do that for for hundreds. Yeah we're going to go up to two thousand eight hundred aren't we so we're going up in
[Ian:] Two hundreds.
[John:] Each two is equal to one hundred, is that right?
[Ian:] Yeah.
[John:] So there's two squares there one hundred.... Okay?... Stop there a minute I know that's the way I started off. Er how are you doing on just going up in hundreds and not two hundreds? Have you checked.
[Ian:] Yeah. About here I was I just had the urge you know
[John:] Yeah.
[Ian:] at twelve
[John:] especially somewhere
[Ian:] to go
[John:] round the ten every happens to everyone they work you know ten twelve is the obvious one to go for the go eight nine ten twelve fourteen and then once you do it you don't realize you've done it and you don't stop.
[Ian:] Yeah.
[John:] Erm
[Ian:] But
[John:] a good a good way is to mark these twos out first yeah, and get them all marked. I know it's a little bit longer but then when you're going up later
[Ian:] You don't
[John:] You're not see you're not thinking in twos you're not thinking I'll go up one and up two squares and okay. So you could carry on along there. Now what's going to happen along the right hand axis we go up to twenty and so we can go along there in twos and it'll just fit on. Okay so then if you'd like to mark that out in twos.... Another way of doing it of course is that as you're
[Ian:] Hang on. Going up in twos every two
[John:] You j right two sorry two squares erm is one engine.
[Ian:] Yeah.
[John:] I should have been a bit a bit clearer about that. Now as we've got... nought one with a bit more room to fit in it it works a lot better. We can just look at them sort of measure them off right there's two... three, four, five okay. So you're using the ruler to go along. So if you want to just... go along there and mark them off.... Any way that you can think of that will help to get these axes right because most of the time it's going to be this sort of thing I mean you said, Do you want me to work it you know use the graph or just work it out from this. It would be easier to work it out from this wouldn't it?
[Ian:] Yeah.
[John:] The so the equations they give you your ability to handle equations means really it would be easier for you to do it without a graph, half the time. But they want you to use the graph and that's what they're giving the marks for. the twenty okay?
[Ian:] mark.
[John:] So zero... yeah zero is where?
[Ian:] eight hundred.
[John:] Okay.
[Ian:] So...
[John:] Right. and four is at?
[Ian:] Twelve hundred....
[John:] Six is at?
[Ian:] Yeah hold on.......
[John:] Okay looking okay so far aren't they.
[Ian:] Yeah.
[John:] Erm so
[Ian:] should I go on
[John:] Yeah carry on with those. Fourteen.... Okay? So
[Ian:] Yeah. Ten
[John:] er did we do six?
[Ian:] Oh no we've got
[John:] Six at fourteen. Eight?
[Ian:] Hang on... eight at... eight at what?
[John:] Eight at
[Ian:] Yeah.
[John:] sixteen hundred.... Ten?
[Ian:] Ten eighteen.... And twenty
[John:] Twenty at two thousand eight hundred.... Okay and they should all be in a straight line.... Looks like it...
[Ian:] Yeah. I'll just and check them....
[John:] Well that's lovely isn't it one beautiful straight line. You expect it to go a little bit just a tiny bit off
[Ian:] Yeah.
[John:] about half a square or so off. Erm a long ruler is a help. So when N is fifteen what's the weight?...
[Ian:] Twenty three hundred.
[John:] Okay.... And when N is nine what's the weight?...
[Ian:] Seventeen....
[John:] Okay. Now what I'd like you to do I didn't realize it'd got that late actually. What I'd like you to do is do this question again now okay?... And erm draw the graph for that work it all out yourself and another one. It's similar to that but it's very different.... This is the this is the question then, W equals eight hundred minus a hundred N. There's a lorry loaded up with a lot of engines, and it goes off dropping them off and as it as you take an engine so N is the number of engines taken off. Now it's not eight hundred minus a hundred N because it starts of with erm twenty engines on already right. Let's say it starts of with just to make it a bit easier for you so it doesn't come like this let's say it starts with fifteen engines on, and what's its weight with fifteen engines on? Two thousand three hundred. Is is that right? Fifteen times a hundred plus the eight hundred yeah two thousand three hundred. Okay. So it's weight is two thousand three hundred minus a hundred N. Yeah I'd like you to draw a graph of that N is the number of engines taken off. Erm you might have you might st start thinking I don't know what he's getting on I don't know what this is all about. What's what's he mean by it what am I supposed to do. Just have a go at producing that graph and from it find out what is W when... four have been taken off... er when... ten have been taken off and when fifteen have been taken off. Okay?
[Ian:] You couldn't give me a few more equations just
[John:] Equations I certainly could I certainly could. Oh now erm you're not at school next week do you
[Ian:] No.
[John:] want a lesson next week? I'm very happy you do one
[Ian:] Yeah.
[John:] if you want. You do? Okay erm Tuesday Wednesday right
[Ian:] What
[John:] I try to keep it the same time.
[Ian:] what do you suggest I start to learn? Any thing I'm not sure n sure of cos I've r I've really ran the school out of paper threes.
[John:] I I I think you're doing very well actually erm
[Ian:] Is there a book I could buy or...
[John:] book. I wouldn't recommend this one it's just this is good for me because it's got sort of
[Ian:] Yeah.
[John:] examples in it. Erm... distance and speed we haven't done distance and speed erm...
[Ian:] I've got that Lett's text book.
[John:] You have have you
[Ian:] Yeah.
[John:] got that handy?
[Ian:] Yeah I've got it here.
[John:] Okay let's have a look at that I'll tell what we'll do I'll I'll suggest a couple of exercises in that for you to work through. He here could just
[Ian:] Yeah there it is.
[John:] Right...
[Ian:] I don't know where that one was that erm sheet hang on stop. No.
[John:] Leave tho leave those in cos they're marking your place there I won't disturb those and let's have a a little look at this useful ones for you to do.
[Ian:] I got that one you remember the geometry one
[John:] Yeah.
[Ian:] I did get that one.
[John:] You're really brilliant at ge really brilliant at geometry I think it's great. And you're really enjoying maths now aren't you.
[Ian:] Yeah.
[John:] Yeah the the graphs it's just a question of doing a few and getting the feel of it and then we'll. So I'll give you some on erm on graphs. You're okay on trig ratios aren't you? Yeah. Here you are drawing graphs. Straight line. Erm... you know what gradient is right. So if you do this formula in equations, rearranging equations erm... put in standard form. exercises.
[Ian:] Yeah under question things and they give you the answers in blue.
[John:] Yeah I was looking for somewhere he doesn't.
[Ian:] No they always give you the
[John:] Okay.... Solving problems. If you read through in this... about equations... okay. Simple linear equations you don't need to do the erm the brackets. So if you start about page fifty nine.... Two point three O and check through that... and just make sure that you're very happy with the stuff that they're doing. And each time you come to the exercise each time they says they say Here you are solve read through it with a bit of paper over it.
[Ian:] Yeah.
[John:] So you're not looking at the answer. Don't look at the answer.
[Ian:] No.
[John:] Look at the question and then just close the book and say right I'm going to try and solve this equation. If you get stuck, uncover one line just to see how they're starting. Don't look at the whole answer and then cover it up again and think right okay that's a start and then try and do all I mean you're not going to have the book to help you in the exam. So
[Ian:] Yeah.
[John:] try and do all of it on your own if you can. Erm if you can't check through the answers there. So you've got lots of stuff on equations all graduated and work through there some with brackets then with fractions. Formulae and equations yeah? And problems on them rearranging them and then through to these to the graphs. Now I'm not sure whether you do non-linear graphs erm don't forget to cover that side up when you're reading it. But if you go through there... to... all the way through to two point three three. And then in two point three three just do the linear... just li linear just means it gives you a straight line. You can skip I mean you can go back to them if you finish this and you want more to do, skip the non-linear or the the not linear they call them right. Skip the not linear.
[Ian:] Has that got anything to do with physics? Not cos like in physics we're doing these not or nor gates like.
[John:] Erm
[Ian:] You know as in
[John:] not really.
[Ian:] really.
[John:] Not really no it's I mean not
[Ian:] It's just that they say if you're doing physics sometimes you know this might help you
[John:] Oh it does help a lot.
[Ian:] you know
[John:] It does help a lot.
[Ian:] They do say that sometimes.
[John:] It it really is true. Erm and then if you go onto drawing graphs from equations okay so skip that bit and just have a look at page sixty eight.... Okay? Linear equations again, solving equations sixty eight, sixty nine, seventy. Have you done inequalities? Less greater then and less that.
[Ian:] Yeah but I'm not I'm like that I'll have
[John:] Okay.
[Ian:] I have a look at it.
[John:] Ok oh okay, erm I was going to say stop at two thirty five. But erm so you work through this lot.
[Ian:] Yeah. Well I'd rather have more to do than have less to cos
[John:] You're really getting through a terrific amount of stuff and you keep asking for more it's great. Really good okay. Erm so read through... inequalities which is two point three five. Right now you may understand those you may think Oh I can't really get the hang of this. Doesn't matter and also two point three six work through that. Pythagoras you won't have any problem on that.
[Ian:] No.
[John:] And your trig two point three seven.
[Ian:] Yeah.
[John:] They sort of they tie together and you'll be using some equations and next time we'll give you som we'll have a look at graphs again just a quick
[Ian:] Yeah.
[John:] sort of I think you've got it now actually
[Ian:] Yeah.
[John:] it's just a question of you practise some.
[Ian:] I think there was one question I'll I'll root it out for next time erm and I was really stunned on it I might be okay on it now.
[John:] Right have a look at that again then
[Ian:] Yeah.
[John:] because there's going to be some question on graphs
[Ian:] Yeah.
[John:] there's always something on graphs. Erm what have I got here oh I know what I've got mustn't pinch our book must I.... What's the date? Is it the first?
[Ian:] Erm the first no it's the thirty first today.
[John:] Thirty first leave a blank cos I forgot to ask him to sign it. Now if you like you can sign all it says is erm it just it's just telling you what it is and saying that you don't mind them typing up the words form this.
[Ian:] Oh.
[John:] They take out anything that can identify you or me erm if you know if I said sort of George or gave an address or a phone number they that would be chopped out. The tapes are erased afterwards what they do is somebody types it all up and erm then they build it all up on a computer.
[Ian:] Yeah.
[John:] How people use this and they put it in the dictionary. So if you're happy for them to use the tape okay. You just just sign there and that's cos they can't use the tapes without your permission even though they're destroyed. [whispering] Erm yeah that's it. Right [] I'm in the middle of painting at the moment as well s I thought it'll be slack cos everyone's breaking up and everyone'll be saying forget and everyone's saying oh yes I do want a lesson. And I said in the holiday, Oh yes I do. [LAUGHTER] So I I just hadn't allowed for it and er m my wife was saying I've got some finishing off as well quite a students finishing off and erm my wife said, put an advert in the paper. I said, Oh don't bother I always I always get people ringing up saying so and so's mother or father says or someone says you've been teaching them and can you take me or can you take my son or daughter and it always works like that. So my said, Ooh you know what if it doesn't you you know you'd have none for next term sort of thing I said well I've got quite a few for next term still. She said put an advert in. I'd just phoned it through [LAUGHTER] and people started ringing up saying [] you know somebody's mother tells me and can you come and do our son because he wants to do this and Yes sure.
[Ian:] [LAUGHTER]
[John:] That's my luck. Don't forget the tape recorder that would be silly.
[Ian:] [LAUGHTER] Yeah. []
[John:] How are we doing? That looks just about finished that okay. I'm not going to have time to get two I'll have a look through this before... Erm you've got the Lett's it's a good one this.
[Ian:] Well it's my sister's you know
[John:] Yeah.
[Ian:] she passed maths first time.
[John:] Good good. Does she does she help you mush with it? Do you talk to her
[Ian:] Yeah she she's I I hardly ever see her but when I do
[John:] How old is she?
[Ian:] Twenty
[John:] Mhm okay.
[Ian:] So I I don't really
[John:] No when when you other things to talk to.
[Ian:] Yeah.
[John:] other things to talk about rather Yeah she'd get so she starts running when she see's you otherwise wouldn't she.
[Ian:] Yeah.
[John:] Right erm I think that stuff that you're doing there will be fine but particularly at this idea of a gradient think in terms of the slope of a hill which is effectively what it's measuring. You know you say, How steep is that hill? well for every hundred you go along you'd go up fifteen. Oh okay I've got some idea. Erm or they might tell you the angle. Just just think about that and just use that graph there If you wanted to work out the angle there, the tan of the angle would be that over that, but only when the scale is the same on both axes. So you can use this the tangent of the angle use your calculator
[Ian:] What's the measurement for gradients? I isn't there one is it just a number.
[John:] It's just it's just a number be because it's one length divided by another.
[Ian:] Yeah.
[John:] So it you went along erm.
[Ian:] So if you had like metres on one side and centimetres on the other
[John:] Right then your gradient
[Ian:] Is that is that what it's
[John:] Your gradient would be would come out sort of all wrong then.
[Ian:] Mm.
[John:] Yeah. So if you're saying if say it was a railway where the gradients are very not very steep at all. And they said it goes up erm ten centimetres every kilometre. virtually level. You'd think, Oh this is a big gradient. If you drew it on your graph it would be quite steep and it would look as if you were going up ten centimetres for every centimetre you went along which is almost vertical. You know so make sure the units are the same and the scale is the same so you're using the same number of squares for each one. If you want to just measure the angle and use the tan. Now usually in the exams they work it so you can't do that cos they give you a graph already drawn and they'll have sort of erm two squares is equal to one unit along the bottom but only one square equals one unit
[Ian:] Yeah.
[John:] sort of going up. So if you do just measure the angle they say, Find the gradient and erm find the the angle. and you can't just measure it. So you'll have to do that. Erm I I I think you're doing really brilliantly.
[Ian:] Yeah.
[John:] I think you're doing very well. you you can tell yourself now that you you're really
[Ian:] Yeah.
[John:] getting a feel for it what you're doing. So if you work through those chapters erm
[Ian:] Yeah.
[John:] and then go back over stuff that you've done earlier that you haven't been able to understand where you've felt, Ooh I don't know what's going on here. erm and now see if you can make sense of it. Look at what you've written and think, Oh I was just fumbling about here I didn't know where I was going. and write it out again the way is should have been. How it should have been there and [tape ends]