PS4H1 | Ag4 | m | (John, age 50, tutor) unspecified |

PS4H2 | Ag1 | m | (Ruben, age 16, school student) unspecified |

- Tape 123301 recorded on 1993-04-14. Locationmerseyside: Allerton, Liverpool ( Private house ) Activity: Tutorial lesson Interactive teaching

John (PS4H1) |
[1] [laugh] Nine? |

Ruben (PS4H2) |
[2] Erm, mm, [...] . |

John (PS4H1) |
[3] Yeah, have you, have you reviewed your revision to see sort of how you're doing and how effective it is, and why it's getting you down and. |

Ruben (PS4H2) |
[4] No. [laugh] |

John (PS4H1) |
[5] No, how |

Ruben (PS4H2) |
[6] Mm. |

John (PS4H1) |
[7] how is it going, just plodding along with |

Ruben (PS4H2) |
[8] Yeah. |

John (PS4H1) |
[9] you know, got more of this, more of this to do. |

Ruben (PS4H2) |
[10] Yeah. |

John (PS4H1) |
[11] Er, is there any of it that you really enjoy, or sort of enjoy a little bit, or think oh this isn't too bad, or. |

Ruben (PS4H2) |
[12] Mm, [laugh] |

John (PS4H1) |
[13] Or don't scream at the thought of? |

Ruben (PS4H2) |
[14] I don't know, it's something, no, I'll just plod through and. |

John (PS4H1) |
[15] What have you been looking at in the maths. |

Ruben (PS4H2) |
[16] Mm. [17] I've just been going through it really, [...] the textbook. [...] |

John (PS4H1) |
[18] And have you looked at some of these bits at the front to tell you how to make it more effective and. |

Ruben (PS4H2) |
[19] Couldn't see it. |

John (PS4H1) |
[20] Right, okay. [21] I mean this is where it starts, with the number system, erm, do you know what any of these terms are? [22] Do you think it's important or interesting to know? |

Ruben (PS4H2) |
[23] Erm. |

John (PS4H1) |
[24] People keep talking about reals and rationals, and, okay, erm. |

Ruben (PS4H2) |
[25] Negatives. [26] Powers. |

John (PS4H1) |
[27] Are you happy with negatives? |

Ruben (PS4H2) |
[28] Yeah. |

John (PS4H1) |
[29] And powers. |

Ruben (PS4H2) |
[30] Roots. |

John (PS4H1) |
[31] Roots? [32] It's only free rationals . |

Ruben (PS4H2) |
[33] Rationals. |

John (PS4H1) |
[34] Rational and irrational, it's just a sort of, the Greeks had a funny term for it, if you can express it as a fraction, think, seventy five over two hundred and ninety seven, then it's a rational number. [35] If you can't, if you can't find a fraction that fits then we just say it's irrational, so can you think of a, any examples? [36] Get numbers that you can't express exactly? |

Ruben (PS4H2) |
[37] You can't express exactly? |

John (PS4H1) |
[38] Mhm. |

Ruben (PS4H2) |
[39] Like nought? |

John (PS4H1) |
[40] Mhm, no, you expressed that very precisely. |

Ruben (PS4H2) |
[41] Erm. |

John (PS4H1) |
[42] What's, what's pi as a fraction? |

Ruben (PS4H2) |
[43] Three point O, as a fraction, one over three point, four. |

John (PS4H1) |
[44] Some people, some books say, oh take it to twenty two over seven or things like that, but it's not, and the approx , what you get off your calculator is only correct to however many digits you've got on your calculator, |

Ruben (PS4H2) |
[45] Mhm. |

John (PS4H1) |
[46] you can never, people have found pi to millions of places, and it's still, they still don't find a pattern where it repeats, so those are the irrationals anyway. [47] You know what a prime number is, and we had a quick look at, we had a look at the primes that even odd, didn't we? [48] Erm, why are prime fa , do you know what prime factors are, and why there useful? |

Ruben (PS4H2) |
[49] Why are they useful? [50] Er, ... I can't remember now. |

John (PS4H1) |
[51] Okay, erm, if you break a number down into its prime factors, then you can look at the prime factors and see whether other numbers would go into it, so if we take sort of say, three sixty, and find some of its prime factors. [52] You won't find any sevens in there, won't be able to, does fourteen go into three sixty, you can have a look at it's a prime factors and say, well no, there isn't a seven there, erm does thirty six go into three sixty one, we're looking for something that threes and twos, appropriate number, three times three, or two times two times three times three would be thirty six, so you have at look how many twos are there in three sixty. [53] Oh, yeah, we can get two out, and we can get two lots of three out as well, so it would go in, and it's pretty obvious it would go in cos it, it's by ten |

Ruben (PS4H2) |
[54] Mm. |

John (PS4H1) |
[55] but, erm, some of the others are not so obvious, [...] would divide in. [56] Erm, now circular measure, did you actually measure a few round things and find pi? |

Ruben (PS4H2) |
[57] Oh, I did, yeah, with you I did, yeah. |

John (PS4H1) |
[58] Mm. |

Ruben (PS4H2) | [...] |

John (PS4H1) |
[59] Yeah. [60] And you remember actually measuring it, and you know what |

Ruben (PS4H2) |
[61] Yeah. |

John (PS4H1) |
[62] pi means, hopefully, it means something to you now . |

Ruben (PS4H2) |
[63] Yeah, it measures the diameter, divided by the circumference. |

John (PS4H1) |
[64] Which is |

Ruben (PS4H2) |
[65] Circumference divided by |

John (PS4H1) |
[66] Right which is bigger, circumference or diameter? |

Ruben (PS4H2) |
[67] Circumference. |

John (PS4H1) |
[68] Right, and what is pi, roughly to the nearest whole number ? |

Ruben (PS4H2) |
[69] Er, about three. |

John (PS4H1) |
[70] About three |

Ruben (PS4H2) |
[71] Yes. |

John (PS4H1) |
[72] so it's got to be the long one, all the way round divided by how far across it is. [73] Er things like that help if you're not sure, you're thinking oh no, especially in an exam |

Ruben (PS4H2) |
[74] Mm. |

John (PS4H1) |
[75] well was it the diameter divided by the circumference or was it the other way round, and you right them on paper and C over D, D over C, doesn't look any different, and you get locked into the problem and you just back off a bit and think of something real, like when you measured it [...] |

Ruben (PS4H2) | [cough] |

John (PS4H1) |
[76] all the way round, that's about three times across and it's got to be that way round, the all the way round, over how far across. [77] Relationships, similar figures, now we we looked at that, and we looked at scale, erm, and I think last time we were looking at things like though, not too long ago we were looking at price of council paper and this sort of stuff, which one shall we buy? [78] Shall we buy that or buy that one |

Ruben (PS4H2) |
[79] Yeah. |

John (PS4H1) |
[80] erm, are you happy with that? |

Ruben (PS4H2) |
[81] Erm, |

John (PS4H1) |
[82] Shall we have a look at it again ? |

Ruben (PS4H2) |
[83] we probably want to about work again . |

John (PS4H1) |
[84] Right we'll have a look at that again, because once you've got it it's pretty straightforward and you can do it fairly quickly and you can do it very confidently |

Ruben (PS4H2) |
[85] Mhm. |

John (PS4H1) |
[86] and if you haven't got it you could be there all day on the exam, urgh urgh, and also you're thinking I'll put it this way up, that one over that |

Ruben (PS4H2) |
[87] Mm. |

John (PS4H1) |
[88] okay, so we'll have a look at that and, obtaining money, well [laugh] |

Ruben (PS4H2) | [laugh] |

John (PS4H1) |
[89] it's easy, you just say I'm the Government, give me your money, |

Ruben (PS4H2) |
[90] Mm. |

John (PS4H1) |
[91] have a poll tax, and let's have fifty per cent V A T, why, make it a hundred per cent and then it's easy for people to calculate. |

Ruben (PS4H2) |
[92] Ha, yeah. |

John (PS4H1) |
[93] If you go and buy a a T V for a hundred and fifty pounds they'd say thank you, a hundred and fifty pounds, plus VAT at a hundred per cent is another hundred and fifty to go to the government |

Ruben (PS4H2) |
[94] Mm. |

John (PS4H1) |
[95] So, yeah. [96] Now what about doing percentages? [97] Like VAT an , are you okay on that ? |

Ruben (PS4H2) |
[98] Yeah, yeah, I'm okay . |

John (PS4H1) |
[99] Good, good, because that that comes up all the time. [100] Geometry. [101] Lines angles and intersections, you were good at that, you were using your compass and doing things okay, and geometry of the circle, I, as I say tend tend to leave that with all the |

Ruben (PS4H2) |
[102] I think I'm alright on actually . |

John (PS4H1) |
[103] that entails, there's thirteen or so circle [...] , transformations comes up quite a bit |

Ruben (PS4H2) |
[104] Mm. |

John (PS4H1) |
[105] and solids and nets, they tend to turn up, right, and what about sets? [106] That's |

Ruben (PS4H2) |
[107] Yeah. |

John (PS4H1) |
[108] statistics and probability. |

Ruben (PS4H2) |
[109] I think I'm alright, because we have covered that all haven't we? [110] With special probabilities, we've covered that |

John (PS4H1) |
[111] Yeah. |

Ruben (PS4H2) |
[112] well. |

John (PS4H1) |
[113] So you can, what what I'm thinking of now is not so much, I've been thinking of stuff that you have done, and that if you sort of spend a little bit of, not much time, maybe ten minutes on each topic, just having a little look at it and finding a problem, this is the sort of problem I was doing a month or two ago, can I do this one? [114] Can I remember how to get started in the main thing? [115] Usually |

Ruben (PS4H2) |
[116] Yeah. |

John (PS4H1) |
[117] if you get started you can, you palm, the pattern comes back and you follow it through and you get the answer, erm, need to do it every, maybe sort of about once a month or so, for each topic, don't let a topic go for about a month without you looking at it for ten minutes or so, and it will be surprising how that little bit of effort keeps it in your mind, so when you come to the exam you don't sit down there and go, oh, it's two months ago, I was doing everyone right, wonderful |

Ruben (PS4H2) |
[118] Mm. |

John (PS4H1) |
[119] now I just, can't think, can't remember |

Ruben (PS4H2) |
[120] Mm. |

John (PS4H1) |
[121] so it's to keep it in your mind, erm. [122] We you |

Ruben (PS4H2) |
[123] Yeah. |

John (PS4H1) |
[124] did quite a few graphs last time, and you're good on algebra, manipulating things, you're good on trig, vectors and matrices, how do you feel about those? |

Ruben (PS4H2) |
[125] Er, [whispering] what's matrices [] ? [126] Vector, oh dear. |

John (PS4H1) |
[127] Erm, you're doing paper three and you do paper three, you do matrices, might even do determinants, it's not that clear [...] about somewhere before, of course you know that. [128] Erm, so scale, proportion, those are the few, few things brought in together, we'll st , we'll start off with just weight, now we'll find sort of weights of things, and then we'll make it more and more difficult |

Ruben (PS4H2) |
[129] Mm. |

John (PS4H1) |
[130] [...] of less easy. |

Ruben (PS4H2) |
[131] Mhm. |

John (PS4H1) |
[132] Less easy as it goes along, so we've got, start by saying loaf of, what do you want to buy at the moment? |

Ruben (PS4H2) |
[133] Er. |

John (PS4H1) |
[134] Sold by weight. |

Ruben (PS4H2) |
[135] Bar of chocolate. |

John (PS4H1) |
[136] Bar of chocolate. [137] Okay. [138] Bar of chocolate, and what sort of size do they come in, sort of hundred gram |

Ruben (PS4H2) |
[139] Yeah. |

John (PS4H1) |
[140] things. [141] Well, right, erm so a three bars of chocolate, there's one in the hundred grams, [...] [sound of tape recorder being struck] will cost you twenty eight pence. |

Ruben (PS4H2) |
[142] So, one. |

John (PS4H1) |
[143] A hundred grams, for one bar, costs you twenty eight pence, right, and then another one that's two hundred grams and it'll cost you fifty pence. [144] You know what this is going to be |

Ruben (PS4H2) |
[145] Yeah. |

John (PS4H1) |
[146] this is going to be easy he thinks, so you've made your mind up before you go to the shop, but unfortunately when you get there there's a, they've got these special hundred gram bars in because they, they haven't got a hundred, there's a big sticker on them saying ten per cent more chocolate, extra bit free, extra ten per cent free, on the smaller bars, so which ones do you buy? |

Ruben (PS4H2) |
[147] How much does, that one extra ten per cent? |

John (PS4H1) |
[148] This, yeah, the one hundred gram ones have an extra ten per cent added on now. |

Ruben (PS4H2) |
[149] Okay. |

John (PS4H1) |
[150] Think as an exercise, work out |

Ruben (PS4H2) | [...] |

John (PS4H1) |
[151] how each of the prices of each one, and you work out which which way round you're going to do it, and then talk to me about it as you're doing it. |

Ruben (PS4H2) |
[152] Mm. [153] First of all put, get that into grams, that's, that's a hundred and ten grams . |

John (PS4H1) |
[154] Right. |

Ruben (PS4H2) |
[155] So the first bar, one ten grams cost twenty eight P |

John (PS4H1) |
[156] Right. |

Ruben (PS4H2) |
[157] and the second one, the second bar's fifty P. [158] So I wanna get that. |

John (PS4H1) |
[159] So just write the second one out again, the second one you get. |

Ruben (PS4H2) |
[160] It's two hundred grams. [161] It's fifty P. |

John (PS4H1) |
[162] So two hundred grams, fifty P, about erm a hundred and ten for twenty eight, which ones easier to work with? |

Ruben (PS4H2) |
[163] The [...] for fifty. |

John (PS4H1) |
[164] Mm. |

Ruben (PS4H2) |
[165] So now you find out how much for one gram. |

John (PS4H1) |
[166] Right. [167] So good rule is work out the easy one first, now they're going to be about the same [...] , erm, otherwise they wouldn't be asking |

Ruben (PS4H2) |
[168] Mm. |

John (PS4H1) |
[169] you, it'd be obvious. [170] Erm, so if you, if you do the easy one, and let's say it comes out to ten pence per gram, and you do the difficult one and it comes to a hundred and three pence per gram, right, or point nought one pence per gram |

Ruben (PS4H2) |
[171] Mm. |

John (PS4H1) |
[172] what's, what would that indicate to you? |

Ruben (PS4H2) |
[173] Er, E, no point in whatever is a lot cheaper. |

John (PS4H1) |
[174] Would it? |

Ruben (PS4H2) |
[175] The first one, nought. |

John (PS4H1) |
[176] I mean, look, okay, look at these two bars of chocolate that you've got, there re , looking at it they're going to be a about the same price, you're not going to find one ten times as dear as the other, are you? |

Ruben (PS4H2) |
[177] Mm. |

John (PS4H1) |
[178] So if one came out at ten pence per |

Ruben (PS4H2) |
[179] Think your first one's probably got wrong, |

John (PS4H1) |
[180] Oh, right, |

Ruben (PS4H2) |
[181] one of them's wrong. |

John (PS4H1) |
[182] Right, you, you think proba , this doesn't sort of seem right to me, and this is sort of developing your common sense approach to maths, which always goes on top of the actual maths, you've done the maths, yeah, you have a little look and it won't, does that make sense? [183] No, whoops, I might have put one on top of the other when it should have been the other way round, so always have this check, er does the answer look, I mean, no, you can't always tell, but sometimes you can, sometimes you can think well those two are about the same price per gram, aren't they? |

Ruben (PS4H2) |
[184] Mm. |

John (PS4H1) |
[185] Okay? [186] So, work out those |

Ruben (PS4H2) |
[187] Mm. |

John (PS4H1) |
[188] and, say, explain the logic of it, why you're doing it the way you're doing it. |

Ruben (PS4H2) |
[189] Well, if you, you want to find out how much per gram, will each bar so how much the twenty eight P, how much, yeah, how many grams, no, we know how many grams were there. |

John (PS4H1) | [...] |

Ruben (PS4H2) |
[190] How many pennies per gram? |

John (PS4H1) |
[191] How many pennies per gram? [192] Good. [193] This is, this is the part where a lot of people get stuck. [194] You can work it out, how many pennies per gram, or you can work it out, how many grams per penny, and it doesn't matter which way round you do it, as long as you know what the answer means, but which way do you prefer? |

Ruben (PS4H2) |
[195] Mm. [196] Er, how many pennies per gr |

John (PS4H1) |
[197] How many pennies per gram, I would prefer that too, so just sort of choose one way and stick to it, I always do it that way, and then you're not confusing yourself doing it different ways. [198] So how many pence would you pay for a gram on that first bar? |

Ruben (PS4H2) |
[199] A hundred and ten divided by twenty eight. |

John (PS4H1) |
[200] Alright, let's look at that, let me just have a little look down here |

Ruben (PS4H2) |
[201] Okay. |

John (PS4H1) |
[202] a hundred and ten divided by twenty eight, you've got this is pence per gram, a hundred and ten grams divided by twenty eight pence, the answer to this is grams per penny, |

Ruben (PS4H2) |
[203] Mm. |

John (PS4H1) |
[204] it was a hundred and ten, wasn't it? [205] Hopefully, |

Ruben (PS4H2) |
[206] Yeah. |

John (PS4H1) |
[207] we've got a hundred grams divided by twenty eight pennies, the answer is whatever it comes to grams per divided by pennies, grams per penny, so that's a good check that you've got it the right way up, before you go and calculate it you just have a little look, what have I got here? [208] I've got grams per penny. [209] You got, and you can write it like that when you write it down, a hundred and ten grams divided by twenty eight P equals whatever the number is grams per penny. [210] So that's a easy way to make sure that you, because that was the first thing you went for now, in the exam you might well put that down and think, oh, hang on, doesn't look right, perhaps it's the other way up |

Ruben (PS4H2) |
[211] Mm. |

John (PS4H1) |
[212] and then start losing confidence, wasting time, if you've got some check, like that, right, okay, so if you like to work that one out, you shouldn't have any problem with the, the second bar, if you can do the first one. |

Ruben (PS4H2) |
[213] Three point nine. [214] Three point nine two. [215] Three point nine grams per penny. |

John (PS4H1) |
[216] That's a [...] leave the three point nine two, because they might be, they may be close, they may not. [217] Okay? |

Ruben (PS4H2) |
[218] The other one's a hundred divided by fifty. |

John (PS4H1) |
[219] Mhm. |

Ruben (PS4H2) |
[220] Four. |

John (PS4H1) |
[221] Right, so there's not much, this looks as if these answers could both be correct because |

Ruben (PS4H2) |
[222] Mm. |

John (PS4H1) |
[223] they're about the same, so when the hundred gram bar has got the extra ten per cent it's it's just just makes it the better bargain, er, they're not losing much on that cos they save on the wrapping, so it probably costs them the same |

Ruben (PS4H2) |
[224] Mm. |

John (PS4H1) |
[225] they do it that way. [226] Now what happens if it gets a bit more complicated, and all it, another way another way of checking this, erm, what would happen |

Ruben (PS4H2) |
[227] To buy. |

John (PS4H1) |
[228] what would happen if you, let's have a look at G, depends on, what would happen if you were still paying fifty pence, but they gave you four hundred, four hundred grams of chocolate, well you get eight grams per penny, this is another test that you got it the right way up, if it were still only two hundred grams, but you payed more money for it, let's say they charged you a pound, I think you'd get less grams per penny. [229] So good ways of checking that you got it right, and then you can go on comfortably with, let's have a look at erm this time you're buying a block of gold, er |

Ruben (PS4H2) |
[230] As you do, yeah. |

John (PS4H1) |
[231] which reminds me, yeah, the last block I got I'll have to take it back, it was the wrong colour, we've got some silver got some, got |

Ruben (PS4H2) |
[232] Yeah, I hate it when that happens, yeah. |

John (PS4H1) |
[233] some silver gold. |

Ruben (PS4H2) |
[234] Ah, it's terrible, innit? [235] Did you have your receipt? |

John (PS4H1) |
[236] No. |

Ruben (PS4H2) |
[237] Oh. [238] That's awful, that is . |

John (PS4H1) |
[239] I went to the, I went to the gold shop when they were shut actually,w to get it in the first place |

Ruben (PS4H2) |
[240] Mm. |

John (PS4H1) |
[241] that's why they wouldn't give me my me give me the money back |

Ruben (PS4H2) |
[242] Ah, of course, yeah. |

John (PS4H1) |
[243] So here's one lump of gold, okay, and that's erm how big shall we make it? |

Ruben (PS4H2) |
[244] That big. |

John (PS4H1) |
[245] A metre? [246] Going to need a truck to carry this home. |

Ruben (PS4H2) |
[247] Yeah, you never have any negatives, you. |

John (PS4H1) |
[248] Mm. |

Ruben (PS4H2) |
[249] Negative metre. [laugh] |

John (PS4H1) |
[250] Negative metres, okay. [251] So it's a a cube one |

Ruben (PS4H2) |
[252] Okay. |

John (PS4H1) |
[253] metre in all directions, |

Ruben (PS4H2) |
[254] Mm. |

John (PS4H1) |
[255] and they're selling them off, gold, gold prices have dropped, you can get a metre, a cubic metre of gold for, say, a million pounds? |

Ruben (PS4H2) |
[256] Yeah. [257] Twenty P and I'll pay it off. |

John (PS4H1) |
[258] For a million pounds, one million. [259] Now there's somebody else next door selling off,se se there is a surplus |

Ruben (PS4H2) |
[260] Can go next door to us. |

John (PS4H1) |
[261] is a surplus of goods, about a ton in good wool, there's a surplus of gold at the moment of course, |

Ruben (PS4H2) |
[262] Ah of course. |

John (PS4H1) |
[263] [...] all selling them off like their payin paying you to take it away. |

Ruben (PS4H2) |
[264] Yeah. [265] A million pounds, I'd call that paying him. |

John (PS4H1) |
[266] Now this bloke, this bloke unfortunately measures it in centimetres, he would, wouldn't he, it could have been worse, he could |

Ruben (PS4H2) |
[267] Yeah, oh yeah. |

John (PS4H1) |
[268] have measured it in inches or something |

Ruben (PS4H2) |
[269] Mm. |

John (PS4H1) |
[270] erm, and erm he sells smaller bits, but they are cheaper, obviously, and his are eighty centimetre cube of gold, okay? [271] And he says, sells his off at a quarter of a million, per cube. |

Ruben (PS4H2) |
[272] Mm. |

John (PS4H1) |
[273] Which one's cheaper, which one would you go for? [274] How do you work it out? |

Ruben (PS4H2) |
[275] I'd go for, I'd go for him, cos I don't like this centimetre part. [276] Okay. [277] So for one metre cubed, |

John (PS4H1) |
[278] You'd [...] this start a new page and do all your [...] of a page. |

Ruben (PS4H2) |
[279] One metre cubed, costs one million pounds, there it is, eighty centimetres, [...] I'm starting one to get, change numbers, such like that to metres, or at centimetres. |

John (PS4H1) |
[280] Good so that's the first thing you've got to do, get these into the same sort of units this is a sort of variation where we had grams and kilograms and things. |

Ruben (PS4H2) |
[281] Erm, thinking ahead, I'd probably turn that to metres. |

John (PS4H1) |
[282] Okay. |

Ruben (PS4H2) |
[283] Three point eight. |

John (PS4H1) |
[284] Right. |

Ruben (PS4H2) |
[285] It's one metre cubed, it's one point eight, erm, er, but, don't know actually, I'd rather have it in centimetres. |

John (PS4H1) |
[286] Hmm, so would I. |

Ruben (PS4H2) |
[287] Yeah. |

John (PS4H1) |
[288] Any day. [289] Two re , why why would you rather have it in centimetres? |

Ruben (PS4H2) |
[290] Erm, then I can, well because dividing point eight by, that'll make it a strange figure |

John (PS4H1) |
[291] Right. [292] So if you can avoid the decimals and avoid the fractions and [recording ends] |