[speaker001:] get this sorted out. [break in recording]
[John:] it tends to stick on, it's never done that, I don't think, yet.
[Chris:] Yeah.
[John:] Now, have you had a chance to look at anything since last time?
[Chris:] I haven't and I've I've meant to, but
[John:] Ah. you know
[Chris:] I've been like mad mad busy w one thing and another.
[John:] Yeah. Oh I know what it's like.
[Chris:] Even of an evening. Well mainly cos of me mum of an evening, cos she's not been too good, so I've been up and down.
[John:] Oh isn't she? Yeah. Yeah it's erm
[Chris:] Can't seem to get a chance.
[John:] It's really
[Chris:] I mean I want to
[John:] Yeah.
[Chris:] to get i and do some of this and do these
[John:] Yeah.
[Chris:] problems,
[John:] Yeah.
[Chris:] it's getting the time to do it.
[John:] But you need to be
[Chris:] Mm.
[John:] When we cover something in the lesson, you need to go over it yourself,
[Chris:] Yeah.
[John:] work out the exercises in it, erm maybe get a little bit stuck, sort it out yourself, and then think, Oh yeah
[Chris:] Yeah.
[John:] I know I see how you do it. And maybe if you're totally stuck, then I can go through it again next time, but
[Chris:] Yeah.
[John:] It's really the more you do
[Chris:] Mm.
[John:] on your own. Because when it
[Chris:] Well
[John:] comes to the [LAUGHTER] exam you're gonna be on your own [].
[Chris:] Yeah. I'll have I mean I'm having er a weekend in Lytham so I might er take stuff up with me, you know for for the odd
[John:] Yeah.
[Chris:] revising. Kids are on the beach and that. So I can work I I can still work through on these ones.
[John:] Okay. You happy with the trig then?
[Chris:] Erm yeah I'd I'd say it's trig and geometry and the bearings, using trig with the bearings is the
[John:] Yeah.
[Chris:] thing that I need to
[John:] Okay.
[Chris:] Cos I know they're the things that seem to
[John:] Okay.
[Chris:] have the big marks on.
[John:] Take a take a bearing on me then, if North is that window.
[Chris:] [cough] Erm North's that window,
[John:] Right yeah, don't forget, go
[Chris:] Yeah.
[John:] clockwise even though it means coming the long way round
[Chris:] Yeah.
[John:] They put it on every question and they all come out of the exam and they say, Oh it was fifty degrees wa No, it was all the way, it was three hundred and ten. Oh. There goes the marks on that one.
[Chris:] Yeah.
[John:] So if you remember nothing else, remember that one.
[Chris:] Yeah.
[John:] to make sure you go clockwise. Okay, have you come across anything else that you've I suppose you haven't had, you haven't been coming across much at all.
[Chris:] No. I mean I had act I wanted to er hopefully have a fortnight off before the exam, to revise, but the way this work'looking, I don't what I'm going.
[John:] Mm.
[Chris:] I'm definitely gonna take a few days off, but I wanted
[John:] Yeah.
[Chris:] to get into it properly. You see I've not really got the same trouble with English, because well English is a written thing isn't it,
[John:] Yeah, and you've been doing that for many many years, English
[Chris:] Yeah.
[John:] haven't you? [LAUGHTER]
[Chris:] Yeah. Erm
[John:] Right so right. So you're okay on graphs, because there's usually a question on graphs which er if you you you know if you're okay on them, pretty easy marks?
[Chris:] Erm not too bad, I mean we've done a The last thing I done at at the night school, was a historogram.
[John:] Okay. Yeah.
[Chris:] Is that is that the right name for it?
[John:] Hist hist histogram. Yeah. statistics.
[Chris:] Erm there's a lot of us starting
[John:] Okay, what does that mean then?
[Chris:] Well I I worked out at school well actually the teacher worked out.
[John:] Is is that.
[Chris:] Erm no only partly finished, it would have
[John:] Okay.
[Chris:] gone to about here.
[John:] What's
[Chris:] Yeah.
[John:] Right. What's the main thing missing from it at the moment?
[Chris:] Erm... well apart from like, the next few columns?
[John:] Mm.
[Chris:] ...
[John:] The bit they give you marks for.
[Chris:] Er
[John:] What's what's this measuring up here?
[Chris:] Oh right, that's erm
[John:] And what's that measuring along there.
[Chris:] I can't remember.
[John:] Right. Now if you do that in real life
[Chris:] Yeah.
[John:] erm you're doing a perhaps you're doing a histogram of some jobs that you've been doing,
[Chris:] Mm.
[John:] to see which ones are giving you the good profits, sort of thing.
[Chris:] Yeah. This this was a weight.
[John:] Right.
[Chris:] Weights and measures.
[John:] Yeah, it can be almost anything, but if
[Chris:] Mm.
[John:] you haven't marked on what it is, erm and then you come make get some interruptions, which you're going to do, and you
[Chris:] Yeah.
[John:] come back to it next week,
[Chris:] Mm. Yeah, what is that?
[John:] What was this? [LAUGHTER] And it's So one of the first thing to do is to get your axes marked so that when you're putting stuff in, you know which is which. Cos you can put these the other way up.
[Chris:] Yeah.
[John:] You know, with almost any graph, you can put them whichever way up you like. So
[Chris:] I didn't do that one.
[John:] No. Decide on suitable scale first. Right. And watch out with the scale on these, on histograms, and on any graphs because, what happens I mean I was watching somebody do this a couple of days ago. They're marking off along here, erm every square is two, say,
[Chris:] Mhm.
[John:] and they go, Two, four, six, eight, twelve [LAUGHTER]
[Chris:] Mm er
[John:] Two, four, six,f around ten
[Chris:] Mm.
[John:] is where most people go wrong. Two, four, six, eight, nine, ten, eleven, twelve, thirteen, fourteen
[Chris:] Yeah. And
[John:] and it's it's rubbish then, it's rubbish because the scale doesn't mean anything. One part of the scale it's double, and the other part it's not.
[Chris:] Yeah.
[John:] And all the results come out all wrong. And it's so s I mean it doesn't need a lo any mathematical skill at all really.
[Chris:] Mm.
[John:] It's just a question of you know, marking off every second square. But you'd be surprised how many people don't.
[Chris:] Yeah.
[John:] they get it wrong. Cos it is a boring thing. And you stop concentrating,
[Chris:] Mm.
[John:] get it wrong. [LAUGHTER]
[Chris:] Yeah.
[John:] So erm mark your axes, work out what your scale is going to be, to fit it all on, I mean that that looks like a nice fit.
[Chris:] Mm.
[John:] Just gone on and a little bit left over. That's fine. Erm if it'd all been down here, you could think, Oh well I can make it twice that, take it out to about here.
[Chris:] Yeah.
[John:] Okay. That looks fine, a good fit. What about erm graphs of functions.
[Chris:] [cough] Erm
[John:] I think we'd better have a look at equations actually.
[Chris:] I've done some Now where is it? I've done some functions.
[John:] Right. Functions.
[Chris:] Yeah. One function... er that that yo you're given a function,
[John:] Right.
[Chris:] You're told that that particular function is equal to that.
[John:] Mm.
[Chris:] And then they ask you to find the function, and do something with it, you have to use that function there, you have to use that in it.
[John:] Mm. Well there's a definition of a function there. [LAUGHTER]
[Chris:] Yeah.
[John:] What's a function?
[Chris:] Er a function is is a set a set sum that you're given. A function of a number.
[John:] Mm.
[Chris:] Er as a end result that number, that's a function but you have to do something with it.
[John:] Er... Mm. Tell me some of the things that happen when you're you're working out a function, what would what would be the the overall picture if someone was going to work out a function?
[Chris:] Erm
[John:] What would you
[Chris:] Well you given the function.
[John:] You're given a function, okay, whatever that is.
[Chris:] Mm.
[John:] It might be a bit like a rabbit or something, it's a function, right. And what would you do with it?
[Chris:] Er I'd store it and use it with whatever I'm asked to use it with.
[John:] So you'd be asked to use it with something. You'd be given some input.
[Chris:] Mm.
[John:] Okay, what would you do with that input?
[Chris:] I'd either, plus it, multiply it, divide it.
[John:] So you'd do some mathematical operations on it.
[Chris:] Mm.
[John:] And is that it or
[Chris:] Either that
[John:] Would that produce an would that produce anything?
[Chris:] It'd produce a different answer.
[John:] And it gives an answer, an output,
[Chris:] Mm.
[John:] okay. So the input comes in, the function, you do something with it,
[Chris:] Mm.
[John:] And you get an output. So that's a sort of overall picture of a function. And what's going inside the function? Can it be anything, or what?
[Chris:] Pretty much it can be anything, it could be in brackets, it could be a minus in brackets,
[John:] Mhm. Okay, I'll give you a I'll give you a number.
[Chris:] And
[John:] Erm three hundred and sixty.
[Chris:] Yeah.
[John:] Right. You give me all its factors. Or a few factors, give me some factors.
[Chris:] Er erm... well er twenty,
[John:] Okay yeah.
[Chris:] five,
[John:] Yeah.
[Chris:] erm... ten,
[John:] Okay. Now is that a function?
[Chris:] Mm?
[John:] Is that a function, what you're doing?... I give you a number and you give me some other numbers that are tied up with it in some way. Is that a function?
[Chris:] Not sure. Possibly.
[John:] Right. It's not.
[Chris:] Yeah.
[John:] Okay. What does your what does this say, for a function? [reading] When you take a number, follow a set of instructions, and end up with [] End up with what?...
[Chris:] Er
[John:] You'd end up with
[Chris:] End up with one answer.
[John:] On answer.
[Chris:] Mhm.
[John:] If you end up with more than one answer, then it's not a function.
[Chris:] Yeah.
[John:] Okay, so that's the that's the big thing about it. It's an input, a set of rules to follow, that will guarantee, that you get one output.
[Chris:] Mhm.
[John:] Okay, erm so suppose I ask you to square the number. Whatever number I give you, you multiply
[Chris:] Mm.
[John:] it by itself. Okay? Three?
[Chris:] Nine.
[John:] Minus four?
[Chris:] Minus twelve.... Twelve.
[John:] Erm yeah, four fours?
[Chris:] Sixteen.
[John:] Okay so it's
[Chris:] Sixteen.
[John:] Right so if I give you minus four, you give me plus sixteen.
[Chris:] Yeah because it's multiplied.
[John:] Right.
[Chris:] Yeah.
[John:] Plus four?
[Chris:] ... Erm
[John:] What would you give me?
[Chris:] Er sixteen.
[John:] Yeah.
[Chris:] Yeah.
[John:] Okay. Is that a function?
[Chris:] Yeah.
[John:] Yeah. Cos you're following the instructions, and you're giving me one answer.
[Chris:] Mhm.
[John:] Now you've given me the same answer for two different inputs, but that's okay. That's fine. That's still a function. But if you're giving me different inputs, different outputs for for one input, then that's not a function.
[Chris:] Mhm.
[John:] So if I say, what's the square root of sixteen?
[Chris:] And I said four.
[John:] You could say, Well it's either four or minus four, which one do you want?
[Chris:] Yeah.
[John:] Two answers, so that's not a function. If I say, Well you'll give me the only ever give me the negative version of the square root, so if I give you sixteen, you give me minus four. If I give you nine, you give me?
[Chris:] Er minus three.
[John:] Right that's a function.
[Chris:] Mm.
[John:] There's only ever one answer. Or I could have been a bit less awkward, and said, you just give me the positive one.
[Chris:] Mm.
[John:] The plus four or the
[Chris:] Yeah.
[John:] the plus three. That would be a function, but a function, there's only one answer to it. thought of as a a recipe, for any number. Okay erm the input need not be a number, it can be all sorts of funny things.
[Chris:] Mm.
[John:] And you get one funny thing as the answer. But we can, at this level, we can restrict it to numbers.
[Chris:] Yeah.
[John:] Erm
[Chris:] It's a few Xs in there and there.
[John:] Right now.
[Chris:] went down to like this kind of stuff here.
[John:] Right.
[Chris:] Twelve over three and
[John:] Okay. Now the notation, do you understand the notation?
[Chris:] Where's the notation?
[John:] Two types of notation it gives here. A and B.
[Chris:] Yeah well erm they're basically different functions different questions.
[John:] Erm those are the same function. But he's given it to you in two different types of notation, and you need to know both. So w how would you read this first on?
[Chris:] X squared plus four
[John:] You haven't read the first bit though.
[Chris:] Oh. [reading] Function X in brackets []
[John:] Mm.
[Chris:] So that x, whatever you use it with, has to be X squared plus four.
[John:] Mm. Okay how would you read this one?
[Chris:] And that one is written out, just a different way. That's got the two dots.
[John:] Yeah.
[Chris:] Right. But he normally sticks to brackets rather than, he's got X is it X squared plus four, it's exactly the same.
[John:] Mm.
[Chris:] He's just wrote it out different. He's used the two dots
[John:] Right. Okay.
[Chris:] and
[John:] This nota this notation, is mapping notation, because amongst other things, a function is a special case of a mapping. We won't go into
[Chris:] Mm.
[John:] mappings, but you can have they're very often non mathematical.
[Chris:] Mhm.
[John:] Erm you have four types of matt of mapping. And two of those types er you can they can possible be functions.
[Chris:] Mm.
[John:] So this is read, [reading] A function F, such that the double dot is just read as, such that, [reading] A function F such that X maps to X squared plus four [].
[Chris:] Mm.
[John:] So that any X any number you give to me, you give me X, I'll g say if I'm the function, if you give me X as the input, I'll give you X squared plus four.
[Chris:] Mhm.
[John:] You give me three, as the input, I'll give you three plus four. Okay. So if you gave me six as the input, what would I do with it?
[Chris:] Six squared plus four.
[John:] Okay, and this is in function notation.
[Chris:] Mhm.
[John:] Where that is what the answer is, F of X.
[Chris:] Mhm.
[John:] F of X is the answer. F of X is what F has done to X.
[Chris:] Yeah.
[John:] Someone gave it X, as its input, and it gave you, as an answer, F of X.
[Chris:] Mhm.
[John:] And F of X in this case will be X squared plus four. Exactly the same mechanism,
[Chris:] Mhm.
[John:] the same rules for that. It's a different notation and you
[Chris:] Mm.
[John:] do need to know both. Both of them mean, take a number, square it and add four. And that's your answer. Okay?
[Chris:] Mhm....
[John:] When you find an answer for the function, at a particular number, erm I would
[Chris:] It's called the evaluation of the function.
[John:] Right. And we'll call it the and we'd have said and You find you find the value of the function, for a particular number, a particular input, a particular value of X. And that's evaluating function. Finding the value of the function. Given that F of X is X squared plus four, work out F of two.
[Chris:] Mhm. That's two times two [cough] is four plus four.
[John:] Okay. And all of these. F of nought.
[Chris:] [cough] That's er nought squared... plus four is four.
[John:] Okay. Right. Now gets interesting, and you need to know what the notation is, when you've got more than one function.
[Chris:] Mm.
[John:] Right?
[Chris:] Yeah.
[John:] So this is where you sort of sort out if you really know what's going on. Three functions. Find F of this, G of that, H of that. And you won't have any problem with that.
[Chris:] Mhm.
[John:] Okay is that okay?
[Chris:] These are them there.
[John:] Let's have a quick look. Erm so on one, A, B and C
[Chris:] This is this is one.
[John:] Right.
[Chris:] There.
[John:] There okay. Erm
[Chris:] [cough] F worked out at a half.
[John:] Problems one.
[Chris:] Oh.
[John:] F of five?
[Chris:] F of five is one. what was it, two X
[John:] Right two times five okay. Right. So all these have been marked and they're okay are they?
[Chris:] Yeah.
[John:] Erm right. Now five
[Chris:] [cough] These ones here though, I was surprised I I got them right.
[John:] Mm?
[Chris:] [LAUGHTER] Because I was like well it wasn't quite gelling, but I just done it anyway and it
[John:] Follow the rule
[Chris:] Yeah.
[John:] tells you. Square it, multiply it by two, subtract, whatever it is, just do it.
[Chris:] Yeah.
[John:] And you get the answer. Erm right.
[Chris:] I think three might be on another page.
[John:] F maps to next prime number greater than X. Find F of seven.
[Chris:] What number's that, A? A on three. Well F of seven was eleven.
[John:] Yeah. Right. F of seven, what's the next prime number, well it's not worth trying the evens, nine obviously isn't prime, so eleven is the next one.
[Chris:] Yeah.
[John:] Okay. F of fourteen? It's not fifteen
[Chris:] That was seventeen.
[John:] Seventeen. Yeah. And F of three, next one up is five.
[Chris:] Yeah.
[John:] Ooh no. No that's erm it's not five.
[Chris:] Why not?
[John:] F of seven is fine, F of fourteen is fine, Right. We need two people for this, to work out this function. We're both doing the same job, we're both the same function, which is, find the next prime number up, from whatever number you're given. Now F of F of
[Chris:] Mhm.
[John:] three.
[Chris:] Yeah.
[John:] You work it from the inside out. What goes into this F of, is whatever came out of that F of.
[Chris:] Yeah.
[John:] So I'll be the inside one,
[Chris:] Mhm.
[John:] the one that works out F of three. Right so, you're the gaffer if you like, you're the outside one. You'll come along and you say, right I will I'm going to You're going to take as your input, whatever I give as output. My input was three.
[Chris:] Yeah.
[John:] And so F of three is five.
[Chris:] Mm.
[John:] And that's what I hand on to you as your input.
[Chris:] Mhm.
[John:] Okay. So you've now got to work out F of five.
[Chris:] Function of five?
[John:] And the function is, find the next prime number that's greater than.
[Chris:] Well don't I just go the next one great than three?
[John:] No because that's what that's not your input, that's my input.
[Chris:] Yeah.
[John:] That's my input.
[Chris:] Mm.
[John:] You're waiting there,
[Chris:] Mm.
[John:] You can't do anything, until I've produced my output. Right I'm that first I'm that function, Mhm. And you're this one, outside the gates if you like,
[Chris:] Yeah.
[John:] outside the brackets.
[Chris:] Mhm.
[John:] Waiting until that one's been worked out.
[Chris:] Oh if that's one's five then they go to the next on e after that, which is seven.
[John:] Seven. Right now... a function of a function there.
[Chris:] Mhm.
[John:] It's the same, it's the same function. I think it's probably easier to see it, when you use a different function. Now we don't he hasn't given you any of those here. Er I'm pretty sure they do give you them. Just check the level
[Chris:] Yeah he hasn't [cough] hasn't
[John:] If he gives you that, if he gives you F of F, then he can give you F of G, and I'm pretty sure that you do need to know that, so I won't bother checking. I'll just give you some on those.
[Chris:] Mhm.
[John:] And you can see what this this is in function notation, which is most commonly used, and it's easier to understand, than the mapping notation, most people find.
[Chris:] Yeah.
[John:] So think of a function, keep it nice and simple, cos you're gonna have to work it [LAUGHTER] out [].
[Chris:] Yeah.
[John:] Okay. Think of a function.
[Chris:] Er X squared...
[John:] F of X equals X squared
[Chris:] plus ten.
[John:] plus ten. Well do you want to make, do you want to make it plus one?
[Chris:] Yeah.
[John:] Plus one might be a bit easier. Okay. And I'll think of a function.... Which is three X minus two. Okay?
[Chris:] Right.
[John:] Okay. Now what I want you to do, is to find F of G of X. And to find G of F of X. Now we'll try a few examples first.
[Chris:] Mm.
[John:] With actual numbers. Before we go into that.
[Chris:] Yeah.
[John:] What's F of... G of... three? How would you work that out, sort of talk about it before you get started. What what are you going to do with that?
[Chris:] Er... m well if if function was three, it'd be
[John:] Function isn't three, input, the number that's going in is three.
[Chris:] Mhm.
[John:] Right.
[Chris:] G three
[John:] G of three.
[Chris:] G of three, G of three squared plus one.
[John:] G of three. So that F has got to wait outside, the brackets, till the stuff inside's been worked out.
[Chris:] Mhm.
[John:] So we'll just leave this down here. Waiting to see
[Chris:] Mhm.
[John:] what it comes to.
[Chris:] Mm.
[John:] It's F of something. It doesn't know what yet.
[Chris:] Mhm.
[John:] Right, so what's G of three?
[Chris:] G of three is er G three squared plus one.
[John:] It's not G three squared.
[Chris:] G of
[John:] It's just No, G of three, is three squared plus on.
[Chris:] Yeah.
[John:] Right. It's very important to be precise and know exactly what you're talking about on these, cos you'll get
[Chris:] Mhm.
[John:] you'll just finish up with rubbish if you don't.
[Chris:] Yeah.
[John:] And it's why they put these in
[Chris:] Mm.
[John:] To m to sort out if you were really on top of it and know what you're doing.
[Chris:] Mm.
[John:] Okay. So now we're looking for F of three squared plus one. And what's No we don't. That's right not right. I should have have checked that before I wrote it in. G of... G of X is three times X...
[Chris:] Yeah Yeah G
[John:] Three times X minus two.
[Chris:] G of X is three ti three X three times X
[John:] So it's three
[Chris:] minus two.
[John:] times three, minus two.
[Chris:] Mhm.
[John:] Okay. Erm let's change that, let's make that four, so we don't get confused with this three that's already in there.
[Chris:] Yeah okay.
[John:] So it's three times X which is three times four.
[Chris:] Mhm. Plus two.
[John:] Okay. Minus two. G of X
[Chris:] Right so it's three fours are twelve take away two is is ten.
[John:] Well okay. That's what we've got so far. Now we want to do F of that. Now what does F of that come to?
[Chris:] The function of that is X squared
[John:] Well that's X. That lot is X.
[Chris:] So it's ten squared.
[John:] So it's three times four minus two, squared
[Chris:] Plus one.
[John:] plus one. So that's... F of G of X.
[Chris:] Mhm.
[John:] Now I'd like you that's F of G of X, to work out G of F of X.
[Chris:] Mhm.
[John:] Er the same X. G of F of three. of four sorry. So... Right. What's that going to come to?
[Chris:] G [cough]
[John:] That's what that's what F of X comes to, and that's what G of F G of X comes to.
[Chris:] Right so first of all I work out what's inside these brackets yeah?
[John:] Yeah.
[Chris:] Leave the G there.
[John:] Good.
[Chris:] Now F of four is four squared plus one.
[John:] Right.
[Chris:] Yeah?
[John:] Don't work ou Yeah that's great. Don't work out what that comes to, leave it as four squared plus one.
[Chris:] Right.
[John:] Okay....
[Chris:] So then... G... G of X er is,...
[John:] Now you you can only write exactly what's on there, in here.
[Chris:] Mhm.
[John:] Because that's what G of that's G of thins number.
[Chris:] Mhm.
[John:] And this is G of the same number. So you don't need the G on it now. When when you went from there to there, the F of four disappeared.
[Chris:] Yeah.
[John:] Cos four squared plus one, is F of four.
[Chris:] Right.
[John:] So when you go from there to there, the G will disappear.
[Chris:] Mhm.
[John:] And the brackets.
[Chris:] Yeah.
[John:] Okay. That will all disappear.
[Chris:] Yeah.
[John:] Hello
[speaker001:] Hi.
[John:] How are you then?
[speaker001:] Fine thanks.
[John:] having a nice rest are you?
[speaker001:] Well [LAUGHTER] I am now now anyway [].
[John:] I mean, you know in general, are you sort of sitting back, you know, letting the house run itself and everyone rushing round.
[speaker001:] Oh well not quite really no.
[John:] You get some troublemakers here sometimes don't you.
[speaker001:] [LAUGHTER]
[John:] Thanks very much for
[speaker001:] Okay.
[John:] the coffee.... So... what would G of seven be, it would be three times seven minus two. But what's G of this lot?
[Chris:] Well is it I'd have to work it out wouldn't I?
[John:] No.
[Chris:] Four fours are sixteen but th that would be seventeen.
[John:] Okay do it like that.
[Chris:] So it'd be three seventeen minus two.
[John:] Right so write it as about here write it as three times seventeen, minus two. Okay, but there's no reason why you can't write it as We're doing G of F of, that was F of, and G of is
[Chris:] G of F of
[John:] That's X so we've got three times, four squared plus one.
[Chris:] Yeah. Minus two.
[John:] Minus two.
[Chris:] Yeah.
[John:] Yeah Okay? Right.
[Chris:] That's I put them just
[John:] That's fine. Now
[Chris:] So it helps to put that in the brackets. It helps to put that in a smaller bracket, when I go three times that, minus that. Yeah.
[John:] Right. G of... F of X is going to be what? It's going to be G of What's F of X?
[Chris:] X squared plus one.
[John:] Right, what do you do with when you're doing G, you do three times it...
[Chris:] Minus two.
[John:] and then subtract two.
[Chris:] Right.
[John:] Right now you worked out F of four.
[Chris:] mhm.
[John:] G you worked out G of F of four. This should be a general thing now. Er G of F of X is equal to that. So if we put four in there, we should get the same answer that you got. Three seventeens minus two, forty nine. Is that right? Looks about it. Yeah.
[Chris:] Mhm.
[John:] Okay. So if we put four in there, that'll be seventeen, three seventeens. Yeah. That works okay, that.
[Chris:] Yeah.
[John:] And G of F of four, would come to forty nine. Now... I'm pretty certain that you do that and it's it's a bit of erm there's qui quite a bit in it. Getting sort of
[Chris:] It does, the functions can see seem quite advanced really don't they.. Or is it just cos it's fresh stuff?
[John:] Er
[Chris:] bit like algebra isn't it?
[John:] But they're not so much Yeah. They're not so much advanced, as they are very very basic.
[Chris:] Mm.
[John:] They're more basic than algebra,
[Chris:] Yeah.
[John:] more basic, and the more basic things are, the more strange and sort of
[Chris:] Mm.
[John:] more awkward to get your head round it. Er just see what they say on functions here..... look in the table. Page eight. Coverage of topics. Topics, relevant to basic level. Higher level, intermediate level. So intermediate, rounding,... simultaneous equations by graph.... They're all on that. want to look at, on the graphs. Bearings, transformations, probability, I can't see functions. Should be right at the beginning. Erm, rounding off. It still ne Yeah. Drawing bearings, transformations, rotations. functions, fractions.
[Chris:] Functions.
[John:] Functions and combinations of functions, at the higher level. N E A, level R. You're doing Q aren't you?
[Chris:] Yeah.
[John:] Are you all doing Q at on your night school course?
[Chris:] Yeah. They might have just threw in a bit er more from paper three.
[John:] Erm
[Chris:] Cos he reckons that paper three does give you some
[John:] Well I I thought, that functions came into it, I must say. this one doesn't give it and er
[Chris:] Is this one er
[John:] Oh this is an eighty eight.... eighty eight. I mean they they can change in a year. Can change in a year.
[Chris:] In this one?
[John:] Er probably the best way to tell, is did he give you a syllabus at some stage?
[Chris:] Yeah. He did.
[John:] Mm. it'll be in here. Cos it's on definitely on the higher level. It's just that I I'm quite sure that you need. See he's given you a function of a function.
[Chris:] Mhm.
[John:] F of F of erm and I think you do need it on that. But erm
[Chris:] See I don't know where he got it from, because he didn't give us th questions out of this book.
[John:] No.
[Chris:] He he'd obviously done them in school
[John:] Yeah.
[Chris:] and brought them in on you know, erm sort of
[John:] Are you reasonably happy with that?
[Chris:] Yeah. I mean at first I didn't know what you mean, but I do now.
[John:] So it's getting used to, for a start, being able to read it.
[Chris:] Mm.
[John:] Cos it's it's hieroglyphics.
[Chris:] Mm yeah.
[John:] Isn't it? It's just
[Chris:] Mm.
[John:] funny squiggles and you've got to sort of like when you first see X squared, and you think, What's that? X with a little two up in the air.
[Chris:] Yeah.
[John:] Erm then you gradually get used to it, and when you read it, you've read it several times, it means something to you then. So this' ll come to mean something. Erm er They're all okay, they're all okay.. He's only given you one of those.... Do do don't bother about that one. Er when do you see him next? Wednesday?
[Chris:] Tues Tuesday.
[John:] Tuesday.... Ask him whether you need to do that C whether you
[Chris:] Mhm.
[John:] need to do that?
[Chris:] Yeah.
[John:] G G of F of X.
[Chris:] Mhm.
[John:] Erm if you do, we'll have a look at it a bit more.
[Chris:] Right.
[John:] Erm I think you're okay on the basic functions. You've got that
[Chris:] Mhm.
[John:] so we'd be better off, spending the time on on graphs.
[Chris:] Now he give me that.
[John:] Three posts on a building site. Er let's have a look. Er one... Right... You okay on number one?
[Chris:] I haven't even read it. [reading] Er the cash price for double glazing the windows was three nine five oh. He decided to pay by hire purchase. Pay a deposit of twenty percent. Calculate the amount of the deposit. [] Well I mean I know that twenty percent is twenty pound in a hundred, but what's a quick way of doing it on the calculator. What is the quick formula.
[John:] Have you got a well there are several way of doing it, but have you got er a percent button? That's one that's probably th simplest. Which you don't tend to have Oh you have got one there on the equal. Erm what's that? Shift and pers shift and equal?
[Chris:] Mm.
[John:] Er let's see. One hundred plus two shift percent. [whispering] Is there an alt or a another funny key that you use instead of? Not that one. [] One hundred erm maybe it'll only work with the times, generally on scientific ones, they don't use the percent
[Chris:] Mm.
[John:] times
[Chris:] Something to do with a hundred times the amount isn't it.
[John:] Two see you've got a percent key marked there, how do you get
[Chris:] Mhm.
[John:] these, upper ones. Oh you put it into a different mode.
[Chris:] Ah yeah
[John:] That's what you do.
[Chris:] Mhm.
[John:] You put this into a different mode, and it's not worth messing about with that at all.
[Chris:] No cos it mess it up
[John:] Because, you put that into a different mode, you can't get back to your normal mode, and your calculator's useless for the rest of the exam.
[Chris:] Yeah.
[John:] So don't use your percent button.
[Chris:] Mhm.
[John:] Erm... right equivalents of percentages and fractions.. What does four percent mean?
[Chris:] Oh four parts of a hundred.
[John:] Okay four percent is a fraction.
[Chris:] Mhm.
[John:] Four out of a hundred. That's it. Simple as that. So if I said, Find four percent of two hundred and forty pounds. Right.
[Chris:] Mhm.
[John:] It's it's four hundredths of don't forget that times is of.
[Chris:] Yeah.
[John:] You don't you don't ever need to know that that is pronounced times.
[Chris:] Mhm
[John:] You can always say of.
[Chris:] Yeah.
[John:] Three of four. Three of four is twelve. Three sets of four, three lots of four, three boxes of four,
[Chris:] Mhm.
[John:] bags of four. Three of four, is twelve. Four of three, is twelve.
[Chris:] Mm.
[John:] Right. Four hundredths of two forty is the same as two forty of four hundredths.
[Chris:] Mhm.
[John:] Doesn't matter which way you round you do it, that of, you can just switch them and get the same answer. Which you know from your your three fours and four
[Chris:] Yeah.
[John:] threes. [cough] so that's all you do. How would you do that on your calculator?
[Chris:] Er two forty times a hundred... times four? No....
[John:] How would you find
[Chris:] Oh two forty over a hundred, times four.
[John:] Right. Yes. Yes. You've got. Are you okay multiplying fractions? Of of-ing fractions?
[Chris:] Yeah. Yeah. Er not a hundre
[John:] Not a hun
[Chris:] Not brilliant No.
[John:] Not a hundred percent that's
[Chris:] yeah.
[John:] a fraction so you must have some idea of fractions. Right.... Easiest thing to do with fractions, is to multiply them.
[Chris:] Mhm.
[John:] It's the easiest thing to do. Cos you do what you think you're suppo you know what you'd expect.
[Chris:] Yeah.
[John:] Is you just multiply the two top ones together.
[Chris:] Yeah.
[John:] Right? And you multiply
[Chris:] Yeah.
[John:] the two bottom ones together..
[Chris:] Right so that's
[John:] So what's Have a guess, what would be erm
[Chris:] So nine hundred, it works out at nine pound odd
[John:] What would two thirds of three quarters... come to roughly. Just sort of thinking about it having a guess.
[Chris:] A half.
[John:] How did you get that? Did you work it out here?
[Chris:] No it just seemed to spring to mind.
[John:] Okay, what would three quarters of two thirds be?
[Chris:] Three quarters of two thirds, be a half.
[John:] Right. Are you happy with that?
[Chris:] Yeah.
[John:] Yeah. Okay. Just just have a quick look at it. In real life instead of squiggles on bits of paper.
[Chris:] Right If we multiply that up it would come out to a half
[John:] Yeah.
[Chris:] wouldn't it. But I never, it just seemed to click as a
[John:] What are we doing. Erm two thirds of three quarters, or three quarters of two thirds.... Right. So how much is there?
[Chris:] Er three quarters.
[John:] How big is that?
[Chris:] That's... that's a third.
[John:] Right and that's also a third. It doesn't look it cos it's
[Chris:] Mm.
[John:] it's a s different colour, but it is the same size as that.
[Chris:] Mhm.
[John:] So we've got two thirds there.
[Chris:] Yeah.
[John:] What would a quarter of that look like? If you shared that two thirds out evenly between four people,
[Chris:] Mm.
[John:] how much would they get each.
[Chris:] About that.
[John:] Great. If you shared out that two thirds right that's that's
[Chris:] Yeah.
[John:] one piece there, two thirds. Share that out between four people, how much would they get each?
[Chris:] Er one of these.
[John:] One of these. Okay?
[Chris:] Mhm.
[John:] So that's one quarter.
[Chris:] Mhm.
[John:] So three quarters, would be three pieces like this.
[Chris:] Mhm.
[John:] It would be one two three.
[Chris:] Yeah.
[John:] That's the half.
[Chris:] Mm.
[John:] Okay. Now what's two thirds of three quarters? Here's your three quarters. What would one third of this be? If we shared that three quarters equally between three people, how much would they get?
[Chris:] Between three people.
[John:] Mhm.
[Chris:] They'd get erm let's see... they'd get that.
[John:] Right. So they'd get they'd get that much.
[Chris:] Mm.
[John:] That's one third of three quarters. So two thirds of three quarters, would be two pieces like that.
[Chris:] mhm.
[John:] Okay? One two. Which is a half.
[Chris:] Mhm.
[John:] So it does work out.
[Chris:] Yeah.
[John:] It always works out with fractions. It's the easiest thing to do with them. Multiply one by the other.
[Chris:] Mhm.
[John:] leave them there. But it's all all this is okay, but it sort of makes your brain ache after a bit and to just see something real that relates to it it's makes it stick in your
[Chris:] Yeah.
[John:] mind.
[Chris:] Mm.
[John:] So... any number, divided by one, is just the same number, so if we
[Chris:] Yeah. So if we got two, four, O multiplied by four, equal to so what it is nine sixty
[John:] Don't don't need to do the equals, you can go straight on to the divide by a hundred.
[Chris:] And if we divide that by one, nought, nought, and then we do the equals, it becomes nine point six.
[John:] Right. Okay. Good.
[Chris:] I say I thought it was about nine pound odd.
[John:] Right. So
[Chris:] So the the formula, [cough]...
[John:] To calculate... what erm
[Chris:] Percentage
[John:] P P percent if you like. Well
[Chris:] Mhm.
[John:] we wo won't use X. P percent of... thirty five pounds, that's thirty five times P over a hundred.
[Chris:] Yeah. That's it? Yeah. Good
[John:] That works, so if you want a twenty
[Chris:] Mhm.
[John:] percent, it's thirty five times twenty and divide by a hundred.
[Chris:] Mhm.
[John:] Some of them you you might do in your head, like you think, Well I know twenty percent is one fifth, so I'll just divide that by five and that's seven pounds.
[Chris:] Mhm.
[John:] Erm so you can do that, you can work out total payments, and subtract that, and what it should have been. What about number two?
[Chris:] [cough] Let's see erm [reading] The diagram above, represents a rectangular lawn, fifteen by ten, with a circular flower bed diameter six. Taking pi as three... Er taking pi as three, calculate the area of the flowerbed. Area of a circle is... pi R squared [].... So it's three times the radius, squared?
[John:] Mhm. Yeah. So calculate the area of the flowerbed. Okay?
[Chris:] Er so the the flower bed has a diameter of six metres. So the radius is three metres.
[John:] Right.
[Chris:] So... pi is three, times three squared.
[John:] Right. Okay?
[Chris:] Well
[John:] Yeah?
[Chris:] Three squared is nine, so it's three nines, which is twenty seven.
[John:] Right. Okay, twenty seven square metres.
[Chris:] Mhm.
[John:] And then what about B?
[Chris:] [reading] Please calculate the area of lawn remaining. [] So we have to take away, twenty seven metres. Square m [break in recording]
[John:] all the way round
[Chris:] Yeah.
[John:] is the perimeter. Area is the short one. That's not all the way round, that's just multiplying one by the other.
[Chris:] Mhm.
[John:] Okay. That gives you a the length by the breadth or if you're working out er painting a ceiling or something,
[Chris:] Mm.
[John:] length by the breadth to work out how may square metres, then you have a look on the can, one can will cover thirty square metres or whatever.
[Chris:] Mm.
[John:] Okay? So that's okay. So you shouldn't be having any problem with those anyway. [reading] In the diagram, a rhombus, each side of length three centimetres, put X in inside. Given that X is less than three centimetres from here mm mm mm mm, indicate the region in which X must lie. [] Mm. Now, any of these geometry ones, have a look, if you can't get it very quickly, just leave it.
[Chris:] Yeah.
[John:] Erm you can come back to it at the end if you've done everything else, but there's something about these that er I think you're one. Erm some people get hooked on geometry, and they, I'm gonna get this one if it kills me.
[Chris:] Yeah.
[John:] It doesn't kill you, but it means you spend an hour on it when you could be doing the rest of your exam.
[Chris:] Yeah. It doesn't help. Yeah.
[John:] Okay?
[Chris:] Er what what they're saying is, P Q P going round this way, P Q R S, is a rhombus. What does it mean a rhombus? That's the name of that shape?
[John:] Yeah.
[Chris:] Yeah. Cos I've not come across that.
[John:] Okay.
[Chris:] I have seen that shape, but not called a rhomb But each side of length is three centimetres. So three, six,
[John:] They will
[Chris:] twelve.
[John:] each of those is three, alright?
[Chris:] Twelve centimetres. For the whole thing.
[John:] The the perimeter would be twelve centimetres
[Chris:] Yeah.
[John:] Yeah.
[Chris:] [reading] A point X lies inside the rhombus. It is given that X is less than three centimetres from P. [] Right?
[John:] Yeah Right.
[Chris:] [reading] And that the distance P X, is greater than the distance R X. []
[John:] Right.
[Chris:] So it's got to be down here. [reading] Indicate clearly, by shading it the diagram, the region in which K must lie, [] down here....
[John:] Clearly shade in the diagram.
[Chris:] Is that worth about. [reading] A point at X lies inside the rhombus. It's given X is less than three centimetres from P. [] So it means it's not Ah so it means it's got to be up the top end.
[John:] But you just said it's down the bottom end?
[Chris:] I did. But I'm reading it again.
[John:] Mhm.
[Chris:] And it says, It's given that X is less than three centimetres from P. Well these are three centimetres long.
[John:] Right.
[Chris:] So if it's less than three centimetres from P it's got to be up this end. And that the distance P X ir greater than the distance R X. Well that does throw me because then
[John:] Right.
[Chris:] Yeah. Because it's like equal distances.
[John:] Okay. [cough] Right. So you get to that stage, and you think forget this.
[Chris:] Mm.
[John:] Right?
[Chris:] Right.
[John:] How many marks for it? Not many. There wouldn't be many for that.
[Chris:] Mm.
[John:] And you could waste an awful lot of time on it.
[Chris:] Wh what was the answer for that?
[John:] Well I think I'll leave you that one.
[Chris:] .
[John:] Erm
[Chris:] Are you taking the mickey?
[John:] I'll give you I give you a clue. Shall I give you a clue?
[Chris:] Er
[John:] No. [LAUGHTER] It's geometry question, what sort of things do you normally use with, when you're doing geometry?
[Chris:] like pi and stuff like that? You know.
[John:] Mm. What sort of instrument did you normally use?
[Chris:] A protractor.
[John:] Yeah. Anything else?
[Chris:] Compass.
[John:] [whispering] That's more like it. []
[Chris:] Mm.
[John:] Erm nearly all of these, geometry questions
[Chris:] Yeah.
[John:] wh where there're constructions, a compass comes into it. So don't spend much time on that. Cos
[Chris:] Mm.
[John:] you've got lots of other work in this that you should be doing.
[Chris:] Yeah. Right.
[John:] But if you're really absolutely cheesed off, then give yourself am maximum of five minutes on that,
[Chris:] Mhm.
[John:] and then leave it and go and do something else.
[Chris:] Yeah.
[John:] And it'll probably come to you. But erm don't get stuck on those in the exam.
[Chris:] Right.
[John:] If you can't do it right away, leave it.
[Chris:] Mhm. Right.
[John:] [reading] Student asks thirty people, how long it had taken them [] And you draw a frequency table, which you've just been doing there.
[Chris:] Mhm.
[John:] Okay? So that should be okay. That would be a good one for you to try on your own.
[Chris:] Yeah.
[John:] Erm right I I would say, just leave that. Just forget it
[Chris:] Yeah.
[John:] Just forget that sort of problem altogether
[Chris:] Mhm.
[John:] because you can really get stuck on them.
[Chris:] Yeah.
[John:] That you'll get you can pretty much guarantee you'll get one, and it'll be almost the same as that. There'll be different numbers, it won't be a football match, it might be weights of pizzas, it might be the [cough] length to french sticks, you know it could be anything. But
[Chris:] Yeah.
[John:] it'll be very similar to that. Same technique. I think you'll also get something like this.... [reading] Using the root which is not drawn to scale, write down the reading running up there, running up there, running up there, giving the total. Find the numbers of litre petrol. [] Now you should be able to do that one.
[Chris:] Mhm.
[John:] Completely on your own. So this i this is a good thing to work through, this one. Now look. [reading] A field is in the shape of a quadrilateral,. Use the scale one to ten. Make an accurate drawing [] Now if you get one like this,
[Chris:] Mm.
[John:] like six, and you will usually get one of these on, do an accurate scale drawing, then go for that.
[Chris:] Yeah.
[John:] They tell you what to do, do an accurate scale drawing. They will often tell you what scale to use as well. Which they do here. And then all you have to do, is measure off something, a diagonal or
[Chris:] Mhm.
[John:] how far is this one from that? And put your answer in. If your diagram is pretty reasonable, you'll get most of the mark. Erm if your diagram's pretty reasonable and you measure the wrong thing, you can still get quite a lot of the marks.
[Chris:] Mm.
[John:] So it's a And it doesn't take all day. It's a good one.
[Chris:] Yeah. So it's a good one to go for. Yeah.
[John:] Right. Now probability. If you don't understand it, just leave it.
[speaker001:] [LAUGHTER]
[Chris:] Yeah. This was the
[John:] Erm
[Chris:] tree thing isn't it?
[John:] Yeah. Erm [reading] Write the appropriate probabilities on the branches of the tree diagram. [] And we'll have a quick glance at that now. See what you make of it. Three blue and one red. And two draws are made, at random in each case. Draw one, a bead is taken from bag A. And it's put in bag B.
[Chris:] Well bag A
[John:] I think that's a bit nasty that.
[Chris:] Yeah. [reading] Bag A contains three blue beads []
[John:] Mhm.
[Chris:] [reading] and one red bead. Bag B contains three blue beads and three red beads. Two draws are made Draw one, a bead is taken from bag A and put in bag B. [] Right. [reading] A bead is taken from bag B. Write the appropriate probability on the branches of the tree diagram. [] Draw one,
[John:] And they won't be giving you a lot of marks for that I don't
[Chris:] Mm.
[John:] think. Erm I think that's... slightly tricky. It's it's unusual.
[Chris:] Yeah.
[John:] Erm
[Chris:] I mean you're obv er to me you'd there's more chance of you getting a blue bead out.
[John:] Well we're not bot too bothered about, is there more chance or less chance? We want to know exactly.
[Chris:] Mm.
[John:] Exactly how much chance. Erm this is this is real this stuff. This is
[Chris:] Mm.
[John:] erm not you know, silly mathematics, it's probability is based on, it's to do with gambling.
[Chris:] Mm.
[John:] It's all about getting your money. Now there was a bloke called Chevalier de Mer. Who was
[Chris:] Sounds French.
[John:] He wrote to Pascal. Was it Pascal? Er yes. Pascal. Erm who was a a mathematician, a very brilliant mathematician. Erm saying I've got this gambling problem, I don't know whether how much it's worth betting on this. You're a mathematician,
[Chris:] Mm.
[John:] you work it out for him. For me. And he he more or less laid down the whole theory of probability, in a few days, Pascal
[Chris:] Yeah.
[John:] did. Erm in the letters that he wrote to this bloke. Who then went on to win quite a lot of money, cos he could
[Chris:] Yeah.
[John:] he knew what the odds were and everyone else didn't. Erm let's say. A pack of cards. [cough] You draw four cards out, one after the other.
[Chris:] Mm.
[John:] Okay? Now it's gonna cost you a pound to play this game. Okay. If you get four aces out, We make sure it's a fresh deck of cards and everything else, and nothing's fixed.
[Chris:] Mm.
[John:] If those cards that you get are four aces, I'll give you a thousand. Okay? a thousand pound. If they're not, then I get your pound. Do you want to play? [LAUGHTER]
[Chris:] No.
[John:] Would you you know, would you play that?
[Chris:] Yeah. I'd probably I'd have a go. Yeah.
[John:] You'd have a go.
[Chris:] Mm.
[John:] Okay. Well
[Chris:] Probably lose me pound like, but I'd have a go.
[John:] You'd probably lose more than that.
[Chris:] Mm.
[John:] Cos you'd keep having, Ah it's only another pound. You've got fifty off me
[Chris:] Yeah.
[John:] so far, I'll have another go, to try and get
[Chris:] Mm.
[John:] that thousand. Well you'd need about nearly round round about a million goes.
[Chris:] Mm.
[John:] You'd think something like three chances in a million of getting four aces. So if someone had come up to you, and said, Well look, you you bet three pound, to have a go and if you get a four aces, I'll give you a million pound, then it's worth it. That's a that's a reasonable that's a fair balanced
[Chris:] Mm.
[John:] deal. That's okay. But usually, it's very very much loaded in favour in favour of the the banker
[Chris:] Yeah.
[John:] the person who's offering you the game, and he knows the odds.
[Chris:] Mm.
[John:] You don't. [cough] You just think, Ooh thousand to one, that sounds good odds. Can't get much better than that.
[Chris:] No.
[John:] But he should be giving you a million to three. [LAUGHTER]
[Chris:] Yeah.
[John:] Which which is a lot more. So we want to know exactly what the odds are, on this one. A blue one or a red one. Draw one. What are the odds. red. I I like writing them writing them on the nodes like that. That's a red one, and that's a blue one. Right. Draw one, it's from bag A. Bag A has got three blue, and one red.
[Chris:] Yeah.
[John:] start off here with three blue and one red.
[Chris:] Mhm.
[John:] What's the chances of getting a red, what's the chances of getting a blue?
[Chris:] Well... there's one chance of getting a red. And there's three chances
[John:] Well mark the probabilities on here. Where they've put the
[Chris:] Oh right.
[John:] Right.
[Chris:] Three blue, one red.
[John:] Now this looks like a good game to bet on, because a probability of one, is an absolute certainty. A probability of three is just totally unheard of.
[Chris:] Mm.
[John:] You can't have a probability of greater than one.
[Chris:] Yeah.
[John:] So this isn't the probability. What's the chances of getting a red?
[Chris:] The chances of getting a red. There's there's one chance.
[John:] One chance in a million?
[Chris:] One chance in three.
[John:] One chance in?
[Chris:] Four.
[John:] Right. So the chance of getting a red, is one in four.
[Chris:] Ah.
[John:] The chance of getting a blue?
[Chris:] Is three in four.
[John:] And the chance of getting either a red or a blue, If I say, Here you are, I'm gonna you've got this bag and it's got three blue and one red in it. And I'm gonna pick one at random, and I'm gonna have this bet with you that I will get either a red or a blue. And what odds are you gonna Are you gonna play?
[Chris:] Yeah. If you've got.
[John:] There's the bag has got, three blue and one red in it.
[Chris:] Yeah.
[John:] I'm gonna take one out without looking.
[Chris:] Mhm.
[John:] And I'm gonna bet you that I'll either get a red one or a blue one.
[Chris:] Mm.
[John:] Now I'll pay you ten pounds to play this game, and I only want a pound if I win.
[Chris:] Yeah. I'd play it but it's possible that the red one will go when you take it out. And there'll only be blue left.
[John:] Well it's a certainty that I'll get either a red or a blue.
[Chris:] Yeah.
[John:] Right.
[Chris:] Mhm.
[John:] Cos there's only a red or a blue in there, so if I
[Chris:] That's right yeah.
[John:] If I say, I'll bet you I'll get either a red or a blue.
[Chris:] Mhm. Yeah.
[John:] I'm I'm onto a good, I can't lose.
[Chris:] Yeah.
[John:] So I'll just keep raking it in won't I?
[Chris:] Mm.
[John:] Because we add up the probabilities, to find out the chances of getting that or that. We get three quarters, add a quarter,
[Chris:] One.
[John:] which is one. A probability of one, is an absolute certainty.
[Chris:] Mhm.
[John:] One out of one. A hundred times out of a hundred, a million times out of a million.
[Chris:] Yeah.
[John:] Cos probability is always expressed as a fraction. Right. Now this is where the tricky bit comes in. Whatever one we get, goes into the next bag.
[Chris:] Mhm.
[John:] So if we did get Three out of four chances we get a blue.
[Chris:] Yeah.
[John:] If we get a blue one, we put it in the bag, and what would the bag cont what would the second bag contain now? Bag B, it had got three blue and three red, so it's now going to have four blue and three red. Okay?
[Chris:] Mhm.
[John:] Looking at this one, if we pulled a red one out of bag A and put it into bag B, bag B would now have three blue, and four red. Take that one next as we're down here. Blue at the top. Now what's the chance of getting a blue? How many are there in the bag?
[Chris:] Three. And four red.
[John:] Four red. So what's the chances of getting a blue one?
[Chris:] Three in seven.
[John:] That's it. Three out of seven. And the chances of getting a red?
[Chris:] Four out of seven.
[John:] Four out of seven. Okay?
[Chris:] Mhm.
[John:] Going back to the beginning again. Three blues on red. Mot likely, three times out of four it'll be a blue. We put it in the second bag, and that's now four blue and three red. So the chance of getting a blue is?
[Chris:] Four out of seven.
[John:] Four out of seven. And the chance of getting a red?
[Chris:] Three out of seven.
[John:] Down here. Three out of seven. Okay?... Right. What's the chance of getting either a blue or a red? On this on this pick?
[Chris:] Erm
[John:] Four sevenths, add three sevenths,... how many sevenths is that?
[Chris:] Seven sevenths are one.
[John:] Which is one. So it's a certainty that we get a blue or red.
[Chris:] Yeah.
[John:] It's just a check that we've got that covered.
[Chris:] Mhm.
[John:] That we have got the right probabilities.
[Chris:] Yeah.
[John:] Same down here, three sevenths add four sevenths, that's okay. Now what's the chance, that we'll get a blue out of the first bag, and a blue out of the second bag?
[Chris:] Erm... Three and four... and four and seven, so it's
[John:] What do you mean by, Three and four and four and seven?
[Chris:] I was gonna add them up to see how
[John:] Were you?
[Chris:] Yeah.
[John:] What happens if I added erm... an eighth to three quarters? Would the three quarters get bigger or less? Bigger or smaller?
[Chris:] Bigger.
[John:] So you're saying, there's more chance of getting the blue out of out of the second, than there is of just
[Chris:] Mhm.
[John:] Right, and your next one, if you put it back, is four out of fifty two. And the next one four out of fifty two. So you're saying the chances of getting four aces, provided you put it back each time and shuffled them again, would be sixteen out of fifty two.
[Chris:] Mm [LAUGHTER] Yeah. No.
[John:] Your o your chances aren't getting better, they're getting a lot worse
[Chris:] Mhm.
[John:] aren't they?
[Chris:] Mhm.
[John:] With each complication.
[Chris:] Yeah.
[John:] Each other unlikely event that's got to happen. There's a good chance we won't even be there.
[Chris:] Mhm.
[John:] We won't there's there's a good chance we don't even get the first blue.
[Chris:] Yeah.
[John:] So having got it, it's not a certainty that we'll get the next one. We've got to combine these two probabilities, by multiplying them.
[Chris:] Mhm.
[John:] So the chance of getting here, is three quarters times four sevenths. Okay?
[Chris:] Yeah.
[John:] Which will come to three sevenths. The chance of getting here, is three quarters times three sevenths.... Now might as well leave them as as twenty eighths. So that'nine twenty eighths there.
[Chris:] Mhm.
[John:] Twelve twenty eighths up here.... This one, the chance of coming down this way, was a quarter. We can only only start off from here,
[Chris:] Mhm.
[John:] if we've come down there. And most of the time we won't. We will have gone up there three times out of four.
[Chris:] Yeah.
[John:] So to even get a chance of picking a a blue, after we've had a red,
[Chris:] Mhm.
[John:] Only that'll one happen one time in in four, on there. So this is going to be a quarter times three sevenths.
[Chris:] Mhm.
[John:] . And that's going to come to three out of twenty eight. And this one, will be what?
[Chris:] Where one one quarter times four sevenths.
[John:] That's it.
[Chris:] Four over twenty eight.
[John:] And they, what do they add up to? Four and three is seven,
[Chris:] Seven
[John:] and twelve is nineteen
[Chris:] So it's six sixteen plus twelve.
[John:] And nine.
[Chris:] twenty eight.
[John:] Right. Okay.
[Chris:] Yeah. Twenty eight twenty eighths is one. That's right.
[John:] Right. It means we've covered them all and we've got the probability correct. Twelve and nine, twenty one, twenty four, twenty eight out of twenty eight. If we'd have added them
[Chris:] mm.
[John:] The way you wanted to do, and then we added all this lot up, we'd find well,w the chance is more than a certainty. More than one, that we'd finish up there. And very very much more than one, that one of these would happen. At each stage in your tree vertically,
[Chris:] Yeah.
[John:] if you add them up, they should come to one. When you work out the probability of arriving at that point. So if we call that point blue, red, and we call this one here, blue, blue. Right? That's a blue followed by a blue. And this one is a blue followed by a red, A red followed by a blue. And a red followed by a red. That's R R. Okay?
[Chris:] Yeah.
[John:] Erm that's a q sorry that's a quick look at it. Now... the reason this is a bit messy, is that at that stage, depended on which one you get out of here, you putting in the second bag. Erm I haven't seen one before like that at this level.
[Chris:] Yeah. He may be giving us harder papers just to
[John:] I suspect that you don't that you don't ne Yeah. I think he is.
[Chris:] As he gears up.
[John:] I think you don't need that bit.... Erm try it again. Do it yourself, that one.
[Chris:] Mhm.
[John:] But don't when you've made your draw,
[Chris:] [cough] Yeah.
[John:] Don't bother putting the ball in the next bag, in bag B. Bag B's got three blue and three red. just leave it like that.
[Chris:] Mhm.
[John:] When you've drawn the first one, just put it back in the first bag and forget about it and go on. And see how you work your tree out.
[Chris:] Yeah.
[John:] Okay. That's erm. But you need you need to be practising a lot of things now.
[Chris:] Mm.
[John:] A lot. Erm... Right.... [reading] Using the distances and bearings, draw a scale diagram and measure things. [] Good one to go for.
[Chris:] Mhm.
[John:] You should get full marks on that and on that other one.
[Chris:] Yeah.
[John:] They're good ones to go for. That one... [LAUGHTER] No way. You know? []
[Chris:] Yeah.
[John:] You can really get tied up in it and you and I'll just try this now. I know I've been a long time on this but I'll I think it's nearly there.
[Chris:] Yeah.
[John:] They can be nearly there for an hour and a half.
[Chris:] Mm.
[John:] And you keep, Oh yeah I'll just No.
[Chris:] Yeah.
[John:] Just leave it. Leave it. And what's Mrs Pink doing? Alright, again, this is the same sort of thing you get. Average speed before the car broke down. average speed between then and then. The length of the time that she spent in Cambridge. That's a good one to look at. Er now he's get Ms and Ss and Ns on this. So that's Midlands and Southern and Northern. That was a Northern was it?
[Chris:] What does he mean by that?
[John:] Northern paper.
[Chris:] Oh right.
[John:] Northern paper, Southern paper, Midlands paper.
[Chris:] Oh yeah.
[John:] Erm... Okay. Yeah I think I think have a go at those, you should be able to. You should be able to do all of those. I would le definitely leave that.
[Chris:] Mhm.
[John:] There won't be a lot of marks on it.
[Chris:] Yeah.
[John:] So you can really just you know
[Chris:] Mm.
[John:] you can just throw the exam away getting stuck
[Chris:] Yeah.
[John:] on one of those. Erm... bearing... You should be fine on that and on that. Now, drawing graphs of functions. Have you done inequalities?... Less X is less than or equal to six? No.
[Chris:] No.
[John:] Cos that is definitely on erm. Mm. You've done drawing graphs of er Y equals
[Chris:] Done vectors
[John:] Y equals two X minus three. Things like that. Draw a graph of that.
[Chris:] No.
[John:] No?
[Chris:] Doesn't ring a bell, no. Not a graph of it, no.
[John:] you have you has he done any drawing graphs?
[Chris:] Yeah he's done some graph work. Erm
[John:] Yeah. Yeah. You'll be given a function like given in that form, Y equals some function of X. And then draw a graph.
[Chris:] There's some sort of graphs here. I d Oh these.
[John:] quick look at that one. Yeah this is it. Yeah. draw a graph of that equals that for that.
[Chris:] No.
[John:] Okay?
[Chris:] Is that a vector? No?
[John:] No. No. Let's have a look. No. It's just a graph of a function which happens to be a straight line. Erm
[Chris:] my lines I think I've possibly got them
[John:] Mm.
[Chris:] .
[John:] Yes. I think so.
[Chris:] Yeah [LAUGHTER]
[John:] I wouldn't have said wrong, but not correct. So X is minus one. Four X plus one is minus three. One... and... five Okay those points are right. So putting those on the graph, you've got minus one, minus three. Minus one right... One one point should be here.
[Chris:] Mhm....
[John:] You've got nought, one, which is there. Okay. I'll just put a ring in pencil, round the ones that are okay. Nought, one, that's okay. Y equals three X plus five, okay. And you're doing, Y equals four X plus one. Oh I see. Yeah. But erm
[Chris:] They're
[John:] Mm. This one... This minus one, three, just isn't shown anywhere.
[Chris:] Yeah.
[John:] Minus one minus three doesn't even go down to minus two.
[Chris:] Yeah. I d I didn't actually complete this.
[John:] No. Okay.
[Chris:] In fact I I've got a feeling this may
[John:] Erm
[Chris:] have been, written down to actually do
[John:] Mm.
[Chris:] on ano separate pieces of graph paper.
[John:] Yeah. That's what I th or well what I was just about to suggest. It would be well worth you doing that, because they give you, you can almost guarantee a question on graphs. They'll give you some
[Chris:] Mm.
[John:] Nice standard functions like this, that you can feel at home with.
[Chris:] Yeah.
[John:] And draw your graph. Erm let's see. If you've got any different, use two colours there, how many have you got? One, two,... and three, Yeah. I would do them on separate ones. Erm If you haven't got graph paper, erm pick a bit of s squared paper
[Chris:] Mhm.
[John:] Just a normal sort of squared exercise book. I haven't got any with me, and I was just looking at graphs for someone else earlier.
[Chris:] Yeah.
[John:] Erm that's I mean I would really recommend that you make the time.
[Chris:] Yeah.
[John:] It's getting
[Chris:] To do that.
[John:] It's getting very very close, and you've got an awful lot to do.
[Chris:] Mm.
[John:] You've got a tremendous amount to for you, not for me to go through with you,
[Chris:] Mm.
[John:] but just for you to do, on what we've done so far,
[Chris:] Mm.
[John:] to consolidate that, before we can move on really.
[Chris:] Yeah. Yeah erm.
[John:] Erm. [LAUGHTER] I can't find time for you,
[Chris:] Yeah.
[John:] you you'll have to make the time somewhere,
[Chris:] Mm.
[John:] pinch the times from something. Erm to do cos you're gonna, you're gonna be struggling if you
[Chris:] Mm.
[John:] if you can't do a lot more work.
[Chris:] Yeah.
[John:] Er how d how do you feel, you're doing with with the maths?
[Chris:] Erm well I feel I like you say, I can do with more more sitting down and doing it myself so it it sinks in.
[John:] Mm. You could do with a lot a lot more. A lot more exercises
[Chris:] Mm.
[John:] and a lot more Because otherwise, you're gonna come to that exam, you're gonna sit down, and you're gonna say, Oh I did something a bit like this with John, I remember that. I understood it at the time.
[Chris:] Yeah.
[John:] It's gone now. Cos you haven't done it yourself,
[Chris:] Yeah.
[John:] on your own, several times, to practise it, to shake it up and down, get all the bits out of it
[Chris:] Mm.
[John:] and sort of store it away
[Chris:] Yeah.
[John:] where you can just pull it in. You think, Yeah, I know what I'm doing with this. Completely in control of it.
[Chris:] Yeah.
[John:] Okay?
[Chris:] Yeah.
[John:] Erm. What more can [LAUGHTER] I say []?
[Chris:] Well
[John:] You you need
[Chris:] I'll
[John:] to You need to have a good go at these. Go through that paper, and
[Chris:] Mhm.
[John:] erm and do those graphs.
[Chris:] Yeah. And I'll have a a good read of this, and try and write down, things that I've got a block about I mean
[John:] Do do more writing and trying things and reading at the moment.
[Chris:] Mhm.
[John:] You can catch up on the I mean you can do those graphs without reading anything else about it.
[Chris:] Yeah.
[John:] Do one on each sheet of paper. Erm Reuben might have some graph paper
[Chris:] I've got gr I've got graph paper.
[John:] Right.
[Chris:] Yeah.
[John:] Okay? Great well use on sheet for each.
[Chris:] Yeah.
[John:] And do some graphs on that. Cos we've got, really we've got a a lot more... that I'd like to cover.
[Chris:] Yeah.
[John:] Erm we've only had a a glimpse really of probability.
[Chris:] Mhm.
[John:] You can guarantee there'll be one question on that.
[Chris:] Yeah.
[John:] Erm you could probably do with revising the statistics bit. Erm you should also be practising, manipulating equations.
[Chris:] Mhm.
[John:] Changing the subject of equ of an equation.
[Chris:] Yeah.
[John:] Yeah. If I give you something like,... now can you remember how to do that sort of thing?... That's that's
[Chris:] Make R the subject.
[John:] Call that three. Two erm... Make Y the subject.... So that you get Y equals something. And again there. R equals something. Now in that one, erm one, this is the easiest one, I thought, Well, better give you some easier ones so you can work into that.
[Chris:] Mhm.
[John:] Erm... [whispering] And make X the subject of that. X equals []... Okay.... No. Have a go at that one first, and work up that way, how's that sound? There's more work in doing these but you you should be sort of practising equations all the time.
[Chris:] Yeah.
[John:] Erm, doing awkward ones that have got fractions in. So that you're keeping up with the fractions. Doing ones that have got percentages in. Doing ones that have got maybe a mixture, of percentages. I mean it was, a couple of years ago, a favourite question, things like, erm, What's twenty five percent of a quarter?
[Chris:] Mhm.
[John:] Er how would you go about that? What's twenty five percent of a quarter?
[Chris:] A quarter of a quarter.
[John:] Right good, okay, and what will that come to?
[Chris:] A quarter... once one is one.
[John:] A quarter of a quarter
[Chris:] Dunno.
[John:] One times one?
[Chris:] One.
[John:] One....
[Chris:] Sixteen.
[John:] Four times, don't forget, four times.
[Chris:] Yeah.
[John:] Er very tempting to add them.
[Chris:] Mhm.
[John:] Especially when you'd rather work with eights than sixteenths.
[Chris:] Yeah.
[John:] Okay?
[Chris:] Yeah.
[John:] It's all this sort of the bias that you've got through how you've worked things, and fiddle about with sixteenths. Erm when you once y once you've got out of the habit of sixteenths and thirty seconds, and things like that, you find that I mean, we found the other week, didn't you, that you were thinking, Oh these millimetres a bit of a pain. He was much
[Chris:] Yeah.
[John:] happier when it was back in the old feet and inches.
[Chris:] Mm.
[John:] But if you try and go back to it, when you haven't done it for years, you realize how complicated it was.
[Chris:] Yeah.
[John:] With the yards, feet and especially all the, How many yards in a a furlong and
[Chris:] Mm.
[John:] a chain, and mile and
[Chris:] Yeah.
[John:] Then nautical mile and everything else.
[Chris:] Mm.
[John:] It's er it's a pain.... That's a paper.. Never mind. That's... paper
[Chris:] Yeah.
[John:] That one was for how old is er... Is that yours?
[Chris:] No.
[John:] Oh that's probability, it's where it starts. functions didn't we.
[Chris:] Yeah.
[John:] And again, these scribbles, I mean my excuse is they're supposed to be like that, but... These won't mean a thing
[Chris:] Mhm.
[John:] in maybe two or three days time.
[Chris:] Yeah.
[John:] But in a d within the next day or so, perhaps over the weekend, if you can make your notes about what this was Put it in your own words, what was happening here. What does F of G of X mean? Erm just put a tiny note on it. erm you were all right on those.
[Chris:] Mm.
[John:] I would ask him about that
[Chris:] Yeah.
[John:] Ask him if you need to know F of G of X and G of F of X.
[Chris:] Mhm.
[John:] I mean I f I I feel that you do. But it wasn't there
[Chris:] Yeah.
[John:] on that syllabus that I've just looked at, so
[Chris:] Mhm.
[John:] don't want to load you up, with
[Chris:] Yeah.
[John:] more stuff if it's not on your syllabus.
[Chris:] Right.
[John:] So that one should be. Rrr. How do you feel about it?
[Chris:] Mm er not too bad, but I feel, you know, I need to
[John:] You you you really
[Chris:] I need to work on it.
[John:] You really do I mean And what's what's the last time
[Chris:] It's remembering the formulas and stuff that's er is the is the major thing, because I know, like in the class, I've not actually had a lesson where I've done bad. But it's memory of er
[John:] If you just have that one lesson, and he explains it to your properly, and he sets you some exercises, on what you've just done, half an hour or an hour ago, you'll get through them.
[Chris:] Mm.
[John:] But if he gives you the s exactly the same numbers even, same exercise, same number
[Chris:] Yeah.
[John:] If he gives you
[Chris:] Yeah.
[John:] a month later, you wouldn't know where to start.
[Chris:] Yeah.
[John:] Unless time from somewhere, and just had I mean, five or six goes, an the same sort of question, until you get to the stage where you just go, Oh yeah, Okay. Bang bang bang bang.
[Chris:] Mm.
[John:] And then you've got to the stage where you don't need to. I mean you know, with other jobs that you do, when you need to practise it, and when you don't.
[Chris:] Yeah.
[John:] Erm if you've got something complicated to wire up or something.
[Chris:] Mm.
[John:] Or work out what I'm doing here, maybe make a little, few little notes to yourself.
[Chris:] Mm.
[John:] I must do that bit and that before that. And make sure this bit is earthed, but that mustn't be earthed and
[Chris:] Yeah.
[John:] You do that same job, half a dozen times,
[Chris:] Mm.
[John:] in the end, you're just whizzing through it, you're thinking about what you're having for your tea. And you're talking to your mate and it's all just happening by magic.
[Chris:] Mm.
[John:] Yeah. And you're not really concentrating very much at all. But it only comes with practice, and it's
[Chris:] Yeah.
[John:] exactly the same with this.
[Chris:] Mm.
[John:] It will you know, you see it done once and do it once yourself. You think, Right I've got it. Come the exam, you will be kicking yourself,
[Chris:] Yeah.
[John:] cos you'll think, I did one of these, when was it?
[Chris:] Mm.
[John:] Oh a couple of months ago I did one of these, now er how do I get into it? Where do I start?
[Chris:] Yeah.
[John:] What's the trick on this one? Hey. So
[Chris:] Yeah. Right
[John:] I don't I mean I know I I'm very busy, I've got stuff that I should have done, months and months ago, that I haven't got round to doing yet, cos there's been all sort of interruptions. So er But I'm the one who suffers, cos I'm not getting round doing it.
[Chris:] Yeah.
[John:] So I can't you know, can't advise you
[Chris:] Mm.
[John:] in how to find time. And you really need do it. I've gone on a bit but you do need
[Chris:] Mm.
[John:] to find yourself, several hours a week.
[Chris:] Mhm.
[John:] Erm as well as just going to night school.
[Chris:] Yeah.
[John:] So it's up to you. Erm I'm you know, I'm quite sure, if you don't, if you don't put a lot of work in, then you will finish up not getting through.
[Chris:] Mm.
[John:] And you just sort of have it all hanging over you and
[Chris:] Yeah.
[John:] do it again next year. So
[Chris:] Mm. How long is there?
[John:] A month and a bit.
[Chris:] Yeah.
[John:] There's a bank holiday coming up now, there's another one the end of May,
[Chris:] Mhm.
[John:] there'll be all sort of interruptions and
[Chris:] Yeah.
[John:] There's there's no time at all. It's gonna fly by and very soon, you'll be sort of strolling into that exam
[Chris:] Yeah.
[John:] thinking, Oh if only I spent a bit more time
[Chris:] Yeah I'll find the time
[John:] Okay [LAUGHTER]
[Chris:] yeah.
[John:] Okay I mean, it's not for
[Chris:] Yeah.
[John:] me. It's for you it's for you.
[Chris:] No no. Yeah. Yeah.
[John:] Erm cos if you don't it's then either, be thinking of, Oh well
[Chris:] Yeah it's a lot of effort going nowhere if I don't
[John:] Exactly it's all wasted, what you have put in has just gone then.
[Chris:] Yeah.
[John:] And you start all over again.
[Chris:] Yeah.
[John:] Okay. Anyway I'll better get off. And erm let you have a a little
[Chris:] Right.
[John:] break from, coming in from work, starting on this. ten past ten again. [LAUGHTER]...
[Chris:] Next Friday then
[John:] Yeah I'll I'll [recording ends]