[speaker001:] Erm... somebody I know has asked me to... er... You know I don't know if you know how they compile dictionaries?
[Andrew:] No.
[John:] Well they... they read stuff and they get tape recordings of people talking and they go through and
[Andrew:] Mhm.
[John:] analyze it and they see
[Andrew:] Mm.
[John:] how words are used.
[Andrew:] Yeah.
[John:] Because the language changes all the time,
[Andrew:] Right.
[John:] and they have to keep them up to date.
[Andrew:] Right.
[John:] And... that's about What what what it is is they've just someone's asked me if I'll just do some recordings of some of the lessons. Also it'll help me to see... er if I'm [tut] doing the lesson properly.
[Andrew:] Oh right, yeah.
[John:] Er it's totally anonymous. It's typed up, they don't use the tape itself, they type
[Andrew:] No.
[John:] it up from it with all sorts of weird accents and things put in,
[Andrew:] Alright.
[John:] and anything that could identify anyone is taken out. So that's that's a bit what it's about.... Erm... it's tied up with... Dictionary, it's and and that kind of D do you know of the project?
[Andrew:] No don't know that.
[John:] Okay it's University and do that one. But they all work more or less the same. Could I turn that off?
[Andrew:] Oh sure yeah. The little red button on the top.
[John:] The little red button on the top?
[Andrew:] Oh green o it's the green one.
[John:] .... So that should be... recording now.
[Andrew:] Spinning away there. [LAUGHTER]
[John:] Yeah. [cough] And then it won't it won't bother us at all will it?... Erm
[Andrew:] Yeah there's that maths paper.
[John:] Okay. Now then part of it is they ask me You don't have to, but they ask me to ask people to sign... erm to say It's just to say that you don't mind your conversation being used. But they won't There'll be nothing in it to identify you.
[Andrew:] Oh sure yeah.
[John:] They take out everything.
[Andrew:] I believe you. [LAUGHTER]
[John:] It's it's very it's all done very professionally in that it's very well controlled. Have you got the paper? Right. Have you had a look through?
[Andrew:] Yeah I had a look er through it just before when I got it.
[John:] Okay.
[Andrew:] It's been stuck in my bag all day....
[John:] Now... when you looked through it... and it wasn't an exam, twenty sixth or something was it?
[speaker001:] Yeah.
[John:] ... Did you think, Oh, of course, oh I could have done that, but
[Andrew:] Yeah.
[John:] any like that?
[Andrew:] Erm... there's a couple yeah....
[John:] So we'll have a look at those... and see if we can see why in the exam...
[Andrew:] [LAUGHTER] Yeah.
[John:] it didn't That was obvious.
[Andrew:] [LAUGHTER].
[John:] Okay. So what have we got?
[Andrew:] And this one yeah.
[John:] Okay.... Erm....... Erm what did you think when you saw it in the exam?
[Andrew:] Erm I just... erm read the question like... and I just... I I I thought it'd be like a triangle, you work it out on the triangle and then I thought no that
[John:] Okay.
[Andrew:] can't be right. Cos it's so much longer, the actual... distance it swings through, dunno if it'd be flat.
[John:] Right. So when you looked at in the in the exam, did you think, Oh, they're not giving me enough information here, or There's some trick I don't know, or what did you think about it?
[Andrew:] I just thought it was erm... I dunno, I ju I j just figured they were trying to get us to draw it draw it into a triangle..
[John:] Well did did you answer the question?
[Andrew:] Er no cos I realized that I was that it was totally wrong so I just gave it a miss and went on to something else.
[John:] So you th so you thought, Ah, draw a triangle, and then when you sort of Did you start it start the question at all?
[Andrew:] Yeah, I drew the triangle on the paper and I realized
[John:] And then you thought, Oh that's wrong.
[Andrew:] No that's
[John:] I mean you didn't go back? So you realized it was wrong?
[Andrew:] Yeah.
[John:] You thought, Oh, it's not a triangle. So, what is it?
[Andrew:] It's like an arc like.
[John:] It's an arc.
[speaker001:] [LAUGHTER]
[John:] Right so you can see now how to do it?
[Andrew:] Yeah.
[John:] And how d how do you it?
[Andrew:] Er you measure get the erm... well measure the whole circumference first,
[John:] [whispering] That's it. []
[Andrew:] and then take that as a proportion of the
[John:] Mhm.
[Andrew:] circumference.
[John:] Good. So you... you're looking at It's not a question of erm... [tut] doing much calculating or anything else, it's just seeing what the problem is, isn't it?
[Andrew:] Yeah.
[John:] If someone got the pendulum, gave it a flick
[Andrew:] Yeah.
[John:] and it went all the way round, no problem there. Two pi R. But it doesn't, it just does Is it forty thirty degrees?
[Andrew:] Mm.
[John:] It's thirty over three sixty. So what was that?... Three marks. So it wasn't a lot on that anyway, but it it it's don't forget that's a good indication. I mean if they're only giving three marks for it you're not going to do... a lot of geometry and all sorts of constructions and... complicated things.
[Andrew:] No.
[John:] It's going to be fairly straightforward. So... without knowing any more... than you knew when you sat the exam,... you could pick up an extra three marks there.
[Andrew:] Yeah.
[John:] Yeah. Just by... not being thrown by the question.
[Andrew:] Mhm.
[John:] Or Er don't forget, part of it especially in an exam,... erm part of solving er any problem... is, Oh, I know how to do this.... Ah, that doesn't work.
[Andrew:] [LAUGHTER].
[John:] And then you go back. You find out what's wrong with what you've done and sort of go off in another direction, or maybe follow the same direction slightly and... veer off. So... don't be put off when your first attempt doesn't work.
[Andrew:] Yeah.
[John:] There's still time in an exam, if you realize soon enough as you did, to have another go. Now what about this one?
[Andrew:] That one. I I figured erm...... [reading] C er s complete accurately below accurately... below the part of the wind that shows wind directions []. So I measured that, it was about five centimetres.
[John:] Right.
[Andrew:] And so I figured that Complete must mean continue on. Er but I did it at a scale like erm one centimetre for o for every hour it's blowing in that direction. So I wrote it along the top of the er paper I was using.
[John:] Mm.
[Andrew:] And I thi and erm... I think west... was seven, and that's only five centimetres, so I extended it [LAUGHTER] another two centimetres [].
[John:] Oh.
[Andrew:] And I thought it was complete like.... And then and then completed the rest of it. But what I should have done... was... measured that,... that would have been about forty nine centimetres, this' d be out maybe
[John:] Right.
[Andrew:] seven millimetres.
[John:] Right. So that was... sort of completely misunderstanding the question,
[Andrew:] Yeah.
[John:] really. You're maki making a... an assumption... that they're obviously going to use a scale of
[Andrew:] Yeah.
[John:] one centimetre to whatever it is,... one. I mean thi this wind force here, er what are they? They go up to Did you do the first part of it?
[Andrew:] Yeah.... Yeah I did that.
[John:] Mm.
[Andrew:] the the eight angles divided by three hundred and sixty so three si three hundred and sixty divided by eight.
[John:] [tut] Erm... well let's have a look, again. The wind blows from the north, okay. [reading] Calculate the size of the interior angle... [] of the regular octagon. Let's have a little bit of paper and have a look at that.... You've Have you had your paper back?
[Andrew:] Erm... y it back today.
[John:] Yeah.
[Andrew:] And er then he took it back of us for some reason.
[John:] [LAUGHTER].
[Andrew:] That's teachers.
[John:] . Not very helpful was it? Okay.... A regular octagon.... Erm... how about if I leave you to do that one?... I'll make mine... a regular... hexagon.
[Andrew:] Oh yeah.
[John:] Okay. So... if you want to draw... a hexagon... erm... how many points equally spaced around the circle? We're going to draw a hexagon inside the circle.
[Andrew:] Yes there's six.
[John:] Six, okay. And where will they come? Zero degrees,... Where's the next one?
[Andrew:] Erm... I see, that erm... This one is... That's sixty degrees it yeah.
[John:] Right. So you're going to divide three sixty into six equal arcs
[Andrew:] I see yeah.
[John:] of sixty. And the next one?
[Andrew:] That's a hundred and twenty.
[John:] Okay. And then?
[Andrew:] A hundred and eighty and erm... then... two hundred and forty.
[John:] Okay.
[Andrew:] And er three hundred.
[John:] Okay. And then... you're back to the three sixty or nought again.
[Andrew:] Oh I see yeah.
[John:] [whispering] Okay. [] So we've got... got a a rule somewhere....
[Andrew:] ...
[John:] my ruler.... I keep... losing my... set squares and rules and things. There's one though. Okay where was the centre of that? [cough]... Right. Now this is a hexagon.... Erm join that up, okay.... and we should finish up with about six sides roughly. Now... while I finish it off, if you'd like to just measure... one of the interior angles that I've already done.... And what's it come to?...
[Andrew:] A hundred and twenty degrees.
[John:] A hundred and twenty. Okay. Now according to your theory that you were applying to that
[Andrew:] Yeah.
[John:] one, you just divide You say it's a six-sided figure, divide three sixty by six...
[Andrew:] Oh right yeah.
[John:] Oops.... What angle does that give, that... that sixty degrees?... Where could you find that angle?
[Andrew:] Erm... Could be... oppo other side there if that was extended.
[John:] ... Okay.
[Andrew:] I think so, if this was this this was extended.
[John:] Oh, right. So if you extended that
[Andrew:] That bit there.
[John:] That would be... [LAUGHTER] That is sixty. Now we've got a very special case here haven't we?
[Andrew:] Yeah.
[John:] Because we've got... sixties all over the place... so maybe it wasn't a brilliant one to try... but... it gives us some indication of what's going on. Where's another?... Another sixty?
[Andrew:] Erm... between these, connected.
[John:] Okay. Any other ones?
[Andrew:] ... No.
[John:] Right. Now that one... is bound to be sixty, the... the angle at the centre, because that was how we made it.
[Andrew:] Yeah.
[John:] Put that down and we thought right draw the circle,... space it out... equally,... six equal angles is what we were doing at the same time as six equal arcs, all at sixty degrees, or all at one sixth of three sixty.
[Andrew:] Right yeah.
[John:] Okay.... Now the octagon.... Could you do that one?... [whispering] Okay. So we'll make a little... centre somewhere. []
[Andrew:] Right....
[John:] Okay.
[Andrew:] Right.
[John:] Okay.
[Andrew:] Erm...
[John:] So what's the angle going to be?... How are you working it out?
[Andrew:] Erm... be... three sixty divided by eight.
[John:] Okay. What does that come to?
[Andrew:] Erm... about...
[John:] About. [LAUGHTER]
[Andrew:] Erm yeah.
[John:] Use your calculator....
[Andrew:] [whispering] So that's three sixty er []
[John:] Could you do three sixty divided by four?
[Andrew:] It's nine yeah so four and a half.
[John:] Okay, so forty five.... Okay. You start from wherever you like.
[Andrew:] start that way up.... Bit easier. [LAUGHTER]
[John:] Good.... So you'll get all your right angles marked, each ninety, and then the halfway points... between
[Andrew:] Yeah.
[John:] each ninety.... This is I mean you wouldn't do this in an exam, you wouldn't draw one and work how you draw it, but by doing it now... erm... you get to know... a regular figure. You get to know sort of how to construct one. I mean now if i said, Make me a a figure that's got ten equal sides, you could do it couldn't you?
[Andrew:] Yeah.
[John:] Or twenty.... So you could construct those.... And then join the... join these points up to the centre.... [whispering] Get that out of your way.... So you'll be able to see. []... Well you know what the angle's going to be at the centre already don't you without...
[Andrew:] Yeah.
[John:] without measuring, cos it because of the way... you constructed it.... It's got to be forty five. So we can then... when we've drawn that,... find out what the other angles are,... and try and work out what will always be true and what will depend on which angle you choose. If we draw a join a few of those up through the centre, just so we can mark the angle.... So you can join any two opposite corners and it'll go through the centre.
[Andrew:] Yeah....
[John:] Erm ...... In fact you might as well do the last one as well.... So... [cough] what angle do you know... absolutely definitely without thinking about it?
[Andrew:] .
[John:] And that is what?
[Andrew:] Forty five.
[John:] That's forty five. That's your three sixty divided by eight.... Okay.
[Andrew:] .
[John:] And you've got lots of lots of triangles that are the same....
[Andrew:] That's forty five right?
[John:] That's forty five. It looks forty five.... Definitely.
[Andrew:] Yeah.
[John:] Erm... how can you tell it's forty five?...
[Andrew:] Well...
[John:] I mean is there any way you could
[Andrew:] It's erm
[John:] prove it? I mean you could measure it maybe and say
[Andrew:] Yeah.
[John:] Okay it comes to forty five. You could draw lots of these figures. You did another one sixty, and it came to sixty.
[Andrew:] Yeah. Well it'd be for the... this angle... if you extended it so it'd... be eight... divided by the three sixty isn't it?
[John:] Erm
[Andrew:] Three sixty divided by eight.
[John:] It looks it looks as if it is. I'm not
[Andrew:] Yeah.
[John:] disputing if it whether it is or it isn't, but how can we show that it is? How can we... say,
[Andrew:] Erm
[John:] Well it always will be?...
[Andrew:] Oh right erm...
[John:] Think about what sort of things add up to one eighty.
[Andrew:] Yeah it'll be all... all the angles on this line....
[John:] Okay.
[Andrew:] Erm
[John:] Let's say if we marked... This is forty five degrees.
[Andrew:] Yeah.
[John:] Okay.... What about these two angles, what do we know about them?
[Andrew:] Oh right they've got to add up to a hundred and eighty
[John:] Right.
[Andrew:] right.
[John:] They're both the same angle for a start, because we've got an isosceles triangle, and forty five plus two X must add up to a hundred and eighty.
[Andrew:] Yeah.
[John:] That one there is also X. So here we've got this angle plus two X.
[Andrew:] Yeah so the two X minus... one eighty is
[John:] [whispering] Okay. []
[Andrew:] the outer one.
[John:] It'll always be that one. So this' ll work for anything. Now,... what was the question? What did they ask?... [reading] A size of the interior angle of the regular octagon. [] What do they mean by the interior angle?
[Andrew:] The angle between these two points.
[John:] Right.... And what should it be?
[Andrew:] Erm... A hundred and... a hundred and thirty five....
[John:] Okay.
[Andrew:] Right I see.
[John:] So you've got the right idea... of dividing it by eight, but you needed to take it on a stage and see what the was. Erm...
[Andrew:] like I divided...
[John:] Yeah.
[Andrew:] divi divided divided by eight then minus one eighty.
[John:] Mm.
[Andrew:] That'd get that angle all the time. Right.
[John:] Yeah. It'll always work. Now the other question they've got is, [reading] What is the angle between the two rectangles? [] Well there's one... rectangle... and... there's the other one at right angles to this one.... What's that angle going to be?...
[Andrew:] It looks like forty five. Erm
[John:] It looks like forty five, and it's almost certainly going to be forty five, and we could probably... work out some reason why it would be forty five....
[Andrew:] Would it be erm... a hundred and thirty five for the interior angle?...
[John:] That's a right angle there, and...
[Andrew:] Yeah.
[John:] that's a right angle. Carry on..
[Andrew:] Erm minus one eighty... to get this angle.
[John:] To get this one, yeah.... And then the same
[Andrew:] Yeah.
[John:] ... the other way round to get that one.
[Andrew:] So... if it was that one, if you could get take that one, it could be anywhere... along the plane.... If you were at a right angle If it If it was two right ang er angle lines like that at right angles
[John:] Right. So you do ninety degrees... one one of your ninety degrees
[Andrew:] Yeah.
[John:] off the... off the one eighty.
[Andrew:] Yeah.
[John:] Okay. So are you happy with that, that you can... if someone gives you someone says Draw er a figure, er a regular polygon with twelve sides, you kn you know how to construct one?
[Andrew:] Yeah.
[John:] And by constructing it you learn... its properties if you like. This is that's that's the obvious one. That's the
[Andrew:] Yeah.
[John:] big thing.... That's always... divide three sixty by N to give you the angle at the centre of the circle. What about these two angles, what's the important point about those then?
[Andrew:] Well add up to erm... plus that would add up to n a hundred and eighty.
[John:] Plus that they add up to a hundred and eighty. And.
[Andrew:] And they're the same.
[John:] And they're.
[Andrew:] isosceles yeah.
[John:] Right. So once you know that, these two are the same... you take that from one eighty,... halve the answer... and it'll give you these angles, and if you've got that ang if you've got these angles,... then you can work out most things.
[Andrew:] .
[John:] Any q any questions they give you based on that, you'll be able to do it. Erm... that way you'll have an understanding of it rather than just saying, Well this bit is always. Would it work for a square?
[Andrew:] Erm
[John:] Without drawing it, try and talk about a square. What would happen?...
[Andrew:] Erm
[John:] How would you work out?... If you d Er just sort of talk it through. Drawing a circle, what would you do?
[Andrew:] Well you'd er mark off
[John:] Mhm.
[Andrew:] every... erm ninety degrees.
[John:] Okay.
[Andrew:] ... And draw through the points.
[John:] You'd draw through the four points.
[Andrew:] work if you d drew a diagonal...
[John:] Okay.
[Andrew:] the centre. You know.... Erm... Be ninety. Ninety, ninety, ninety.
[John:] And these angles... would be?... What would the angle
[Andrew:] Erm
[John:] here be?
[Andrew:] Forty five.
[John:] They'd be forty five.
[Andrew:] Isosceles yeah.
[John:] So the interior angle.
[Andrew:] Would be ninety.
[John:] Would be ninety.
[Andrew:] Yeah.
[John:] And the exterior angle... would also be
[Andrew:] Would be forty five.
[John:] would also be ninety.
[Andrew:] Oh yeah it would. Yeah.
[John:] Right? In a square.
[Andrew:] I see it.
[John:] So it would work with a square. Erm... would it work with a triangle?
[Andrew:] Erm... three points... the centre point....
[John:] [whispering] Right. [] So what would the angle at the centre be?
[Andrew:] It'd be erm... twelve.... Hundred and twenty degrees yeah.
[John:] Hundred and twenty. Okay....
[Andrew:] Yes but it wou cos if you had the centre point... up to the top of these two that'd be hundred and twenty, hundred and twenty, hundred and twenty.
[John:] so if someone said like Do a a hundred-and-eighty-sided figure...
[Andrew:] [LAUGHTER] Yeah.
[John:] it it'd just look like a circle. But you could do it. So you're happy with polygons?... Okay. So that's one if it comes up again...
[Andrew:] Definitely.
[John:] No problem on that.
[Andrew:] Yeah.
[John:] So you've obviously dropped a few marks on that.
[Andrew:] Yeah.
[John:] Erm... probably on this and on this one. The big thing there is... there's no rule in life that says... you know if someone says Five them you must draw five centimetres.
[Andrew:] Yeah.
[John:] This might have been... five knots, five miles an hour,
[Andrew:] Yeah.
[John:] decide well we'll have five inches. Erm it... is likely that it's not a linear scale anyway. It's possibly the Beaufort Scale
[Andrew:] Yeah.
[John:] which is a sort of logarithmic scale.
[Andrew:] Yeah.
[John:] Erm so... don't make don't make too many assumptions.... carry on with symmetry?
[Andrew:] Yeah. I got all of er for them more.
[John:] Okay. So let's let's have a look at the others. Now you tell me which ones you think were... an absolute doddle. A giveaway.
[Andrew:] This was... the ones here.
[John:] Mhm.
[Andrew:] [reading] Measure the size of the angle compass. []
[John:] Okay. Did you
[Andrew:] Erm protractor sorry.
[John:] did you use that or
[Andrew:] No cos I didn't have it at the time when I did that.
[John:] Would it have been an advantage to use the?
[Andrew:] Erm... well er yes I thought cos I'm just measuring the
[John:] Yeah. It depends what you're used to. When you get used
[Andrew:] Aha.
[John:] to this one, you'll find it so much easier.
[Andrew:] Yeah.
[John:] Erm... it's a bit off-putting at the start because it's got the two scales on.
[Andrew:] Yeah.
[John:] Make sure you're reading the right one all the time.... [whispering] Okay. []... So no problem there.... You think you got Did you did you see which ones you got full marks on?
[Andrew:] Erm... yeah. I didn't do this I d I didn't quite finish this one cos we only had... erm... one and a half hour to do the two-hour paper like.
[John:] [sucks teeth] Okay. And there were... how many marks on that then?
[Andrew:] Erm... erm eleven marks like.
[John:] Mm.
[Andrew:] Mm.
[John:] So... at the start of the paper,... have a quick whiz through, see where the marks are,
[Andrew:] Do them first, yeah. [LAUGHTER]
[John:] and think, Well I'm going to Not necessarily do them first. I mean you can whi you could go through and you can pick up lots of ones and twos that build up. You're probably going to do them anyway, and they're good to... get you into it. So you're thinking, Oh this hasn't been an exam, these are dead easy, to just play yourself in. Erm... but don't like the
[Andrew:] Mm.
[John:] ones the or others, if you're getting stuck on them, don't miss something like this that you could do and get full marks.
[Andrew:] Yeah.
[John:] Er you know makes a big difference to your final... grade doesn't it?... [whispering] Right. [] Any problems with these?
[Andrew:] Erm no. They they they were they were straightforward, just...
[John:] Okay. How did you do this one?
[Andrew:] Erm I drew a... I drew a erm Venn diagram.
[John:] Right.
[Andrew:] Put my twenty one in the... Whimby circle or Whitby circle, and the thirteen in the erm Scarborough circle.... Put X in the middle, and twenty one minus X.
[John:] Okay. Just just show me how you did that one.... So... a trip to Whitby and a trip to Scarborough.... You've got thirty pupils interested in one or both trips....
[Andrew:] Right....
[John:] Twenty one interested
[Andrew:] .
[John:] in the Whitby... and thirteen... interested in the one to Scarborough.
[Andrew:] So there's an X in the middle.
[John:] How many interested in both of these trips?...
[Andrew:] Erm...
[John:] Now okay. You've got... thirty down there, that's good,... X in the middle, twenty one minus X, thirteen minus X.
[Andrew:] Yeah.
[John:] Okay.
[Andrew:] So...
[John:] So when you get to that point you you've more or
[Andrew:] Yeah.
[John:] less solved the problem haven't you. If you can if you can handle simple equations, you get the answer.... Like most of these, the key
[Andrew:] Mm.
[John:] is what should we call X. Once you've
[Andrew:] Yeah.
[John:] ... sorted that out then you follow your system.... Okay. And does that does that work out? How many would you have in here then?
[Andrew:] Yes you'd have erm... seventeen.
[John:] So you've got seventeen in there, seventeen in Whitby.
[Andrew:] ... And eleven....
[John:] Okay.
[Andrew:] And then erm... four interested in the both trips so...
[John:] What does that come to?...
[Andrew:] Erm erm... it's twelve.... Oh right that's not right is it, no. Erm...
[John:] Mm.... Okay so we've got eleven and four is fifteen, and seventeen... is thirty two.... It says there were thirty pupils.
[Andrew:] Yeah.
[John:] Mm. Now did you see your marks on this one?
[Andrew:] Erm no. I d I didn't er not on that one.
[John:] No that's okay. So there's another one where probably... you can pull up your marks quite easily.
[Andrew:] Yeah.
[John:] Cos you know you know the method, you know what to do, it's just a question of sorting it out. You know a little bit of extra... thinking about it. So one of the points... that sort of comes up is you haven't used the thirty at all have you?... Now did they give you that and it's something you don't need or do you think maybe it's something you do need?
[Andrew:] Er.... I just didn't know how to apply it.
[John:] Mm. Okay.... Is this basic idea okay? We've got X... pupils interested in both, so we've got... twenty one minus X in that and... thirteen want to go to Scarborough, okay? Thirteen minus X in there. Now... where did this come from? Twenty one minus thirteen equals two X.
[Andrew:] Well you've got your X there, you've got your two Xs there, so you subtract them... and you've got
[John:] Mm.
[Andrew:] and then take the... halve.
[John:] Well th you've got an equation, okay. Twenty one minus thirteen equals two X. Now where did it where did it come from? Show me where each term from so you could Thanks very much.
[Andrew:] Cheers Dad.
[speaker001:] Put away your er luggage....
[Andrew:] . Er cheers.
[speaker001:] [whispering] Scattered about. [] [LAUGHTER]...
[Andrew:] Cheers.
[speaker001:] Right. Okay.
[John:] Thanks very much.
[speaker001:] .
[Andrew:] S erm... so I had to subtract these two....
[John:] Mm.... Try and explain it in terms of say... let's get all these kids out into the school yard,
[Andrew:] Yeah.
[John:] and draw two big circles that intersect,
[Andrew:] Yeah.
[John:] and say, Now we want to sort you out, see who's going where, so we can organize the coach now, and if you're interested in Scarborough go and stand in that circle,Whit Whitby there, interested in both then stand where the two circles intersect.
[Andrew:] Yeah.
[John:] Okay. Now take it from there. What will you what how can you work out...
[Andrew:] Well erm... just count how many's erm...
[John:] I mean two teachers come back and one of them says, Well I've counted all those who are going to Whitby. and the other one says, I've counted all those who are going to Scarborough,
[Andrew:] Mhm.
[John:] ... and er nobody... remembered to count the ones in the middle.... But we do know there were thirty pupils, which hasn't Has that appeared in your equation anywhere?
[Andrew:] No not really. [LAUGHTER]
[John:] No. [sucks teeth]... Now erm I still don't really understand where you got this from....
[Andrew:] Erm...
[John:] Why did you you do twenty one minus thirteen for a start?
[Andrew:] Well I had the the twenty one and thirteen in the two groups... so... subtract those two and have twice the amount of X I thought I would have....
[John:] Erm you you seem to be equating the groups. Okay. We won't go into... where you go that from... cos I think it sounds a bit like sort of clutching at straws really.
[Andrew:] Yeah.
[John:] I'm going to get an equation out of this. You you've you've started off brilliantly. You've got marked on you know exactly how many are going to Whitby,... right, twenty one minus X, how many are going to both, and how many are going to Scarborough.... So what could you find out from that?... That ties up with this. That ties up with.
[Andrew:] You could er subtract the number going to Whitby from thirty.
[John:] You could.
[Andrew:] Erm
[John:] You could do something a lot
[Andrew:] And then
[John:] simpler. A lot simpler.... How many pupils were there altogether?
[Andrew:] There were thirty pupils so
[John:] And how many are there standing there in the playground?
[Andrew:] There's thirty.
[John:] Right?
[Andrew:] Right.
[John:] So... an equation?
[Andrew:] Yeah.
[John:] ... Count how many you've got in here. We've got twenty one minus
[Andrew:] So twenty one minus X plus thirteen minus X is equal to thirty.
[John:] And those in the middle as well.
[Andrew:] But oh twenty one minus X plus... X... plus thirteen minus X is thirty.
[John:] Right. So we just right that down, that's the total. So we've got twenty one minus X... those are the ones going to Whitby, and we've got X going to both, and we've got thirteen minus X, and that's the lot. Now assume that this is assuming they all voted. They all said, Yes we do want to go.
[Andrew:] Yeah.
[John:] Erm it would have been better if they'd given you some information to say, They all voted for at least one
[Andrew:] Yeah.
[John:] trip, or there were four who went nowhere. Okay. So....
[Andrew:] Erm...
[John:] What does that give you?...
[Andrew:] So it'd be erm... thirty four minus two X plus X... erm......
[John:] Okay.
[Andrew:] Er X
[John:] So how do you just I mean rather than do it in your head, especially in an exam, just what you're going to do, put the Add X to both sides say....
[Andrew:] Erm...
[John:] If you add X to minus X
[Andrew:] .... Oh it's three yeah.
[John:] Right. Let's have a little look.
[Andrew:] Thirty four equals thirty.
[John:] Right. If we just... take it from there. Thirty four minus X equals thirty. Now as soon as you've got a minus X and you're trying to take it over to one side, and you're trying to bring the other one over, there's a good chance that something's going to go wrong. So why not just follow a nice simple system. Thirty four minus X equals thirty. Okay. Add X to each side.... They go out.... Subtract thirty from both sides.... Thirty four minus thirty equals thirty minus thirty add X.... Okay. So X is equal to?
[Andrew:] Four.
[John:] [whispering] Which... is what you've got there. [] And
[Andrew:] Yeah.
[John:] Now does it add up?... It should do this time if your if... Is this right? We've got Let's check that. got the same answer still.... Twenty one... minus X If I give you some numbers to add up. If I said... twenty seven... add... thirty five add forty nine... add seventeen, would you add them up like that?... Or might you put them one under another?
[Andrew:] Yeah I'd probably put the... numbers
[John:] Okay. So
[Andrew:] which are easier to
[John:] Yes.
[Andrew:] fit ea slip easier.
[John:] Twenty one minus X... plus X... plus thirteen minus X. Tot those up.... Plus X and a minus X go out. That gives minus X.... Thirty four equals thirty. Erm... there may be a simpler way of doing it. Now is this right? Cos we're still getting X equals four.... Add the X to each side,... take the thirty off each side we've got X equals four. So what's going wrong here? Seventeen... and Ah.
[Andrew:] Oh sorry nine yeah. Nine yeah. [LAUGHTER]
[John:] [cough] Because that comes to because that comes to nine.....
[Andrew:] Yeah that's right.
[John:] And that comes to
[Andrew:] . [LAUGHTER]
[John:] that comes to thirty. Okay. So it does come to thirty. Erm... the way you were doing it will sometimes give the right answer but the easiest way
[Andrew:] Yeah.
[John:] ... is to... make sure every box is marked in there. Let's do a slightly more... complicated version. Erm... scribble on here. Now let's sort this out. Erm let's see... Okay. [whispering] Let's see if those... erm... Right. []... This time... [whispering] Let's take that off. []... This time there are twenty nine in the class.... So you've got twenty nine students in the class and... fifteen want to go to Whitby,... and... eight... want to go to Scarborough,... and three little piggies want to stay at home. Three of them don't want to go anywhere.... Twenty nine in the class so what I'd like to know is... erm draw the Venn diagram and find out... the numbers of students in every section of the diagram.... And then if you can do that you can do any... problem of this sort can't you?
[Andrew:] Yeah I suppose so. [LAUGHTER]... three.... Right.
[John:] Okay. So... where are you going to start?
[Andrew:] Yeah but this Right.... That doesn't add up to twenty nine, does it?
[John:] It doesn't add up to twenty nine.
[Andrew:] Eleven....
[John:] Erm
[Andrew:] Twenty six.
[John:] Twenty six. Okay. Well spotted.... I probably mean nineteen or something. Let's have a look. Erm... Erm... Okay. Seven seven don't want to go anywhere.
[Andrew:] Right.
[John:] Seven don't want to go anywhere....
[Andrew:] Right....
[John:] Okay. That's great.
[Andrew:] Erm... that's
[John:] So you've got the various... sections of the diagram. You've got... the Whitby lot, the Whitby and Scarborough, Scarborough and the don't want to go anywhere.... When when you when you add them up... just... if you just put them under each other, it makes it very easy.... So keep the numbers
[Andrew:] That goes there.
[John:] that's it keep the numbers under the numbers and the Xs under the Xs....
[Andrew:] Okay it's twenty nine right.... So an X plus er...
[John:] So X equals one. Okay. What will that give us?... That's
[Andrew:] Fourteen and seven is twenty one.
[John:] Fourteen [tape change] that was sent to me by the the people who want the stuff. So [LAUGHTER]
[Andrew:] [LAUGHTER] Oh right yeah. []
[John:] . Okay what does that come to?
[Andrew:] Er twenty twenty nine
[John:] Twenty nine. Now you might think, Erm oh a bit finickity... saying do it this way, but it's a good system. Erm... also when you're when you're adding up if you get this... fifteen minus X, right, eight minus X and you're putting
[Andrew:] You just cross those.
[John:] the numbers underneath each other and the Xs underneath each other, plus X, and then the plus sign
[Andrew:] X.
[John:] well that goes with the numbers. Yeah? Then when you add them up
[Andrew:] You don't get confused. So I can just...
[John:] Mm.
[Andrew:] take these two off.
[John:] Because there are
[Andrew:] Mm.
[John:] so many, it's very very easy to do.
[Andrew:] Yeah.
[John:] Lots of people'll finish up with, Oh that's a three X and they'll put plus three.
[Andrew:] Mhm.
[John:] And this way it's more obvious what's going on. There's going to be a minus X left over there, and then you can add those up.... Thirty minus X equals, how many were there, twenty nine. And it more or less does itself.
[Andrew:] Yeah.
[John:] So... as I said, if I give you a list of numbers, say two- or three- digit numbers
[Andrew:] Yeah.
[John:] in a long line,... [sucks teeth] horizontally, you'd think, Oh I'd like to put these vertically
[Andrew:] Yeah.
[John:] and do a nice little adding up on it. So why not do it with Xs, because there's more chance of an error. Now why it's a good idea to do it that way
[Andrew:] So I don't in me head like and mixed up.
[John:] Again in an exam, there's more chance that you make a little slip, because... there's little bit of extra edge.
[Andrew:] nerves like. [LAUGHTER]
[John:] A similar one. [cough] Erm this time everyone's at Wembley... and you're asking people,... Right, if you draw this time... we draw three big circles in the middle of Wembley,... okay,... and ask people to come and... say which football team they think is a good one. So we've got Liverpool, Everton and Tranmere Rovers. So they say The instructions to the crowd are, If you think... Liverpool is a a good football team, stand anywhere in that circle, Everton in that circle, Tranmere Rovers, stand in this one. And we get something like this. Erm... and let's say there are a few a few awkward ones here er... who don't think any of them...
[Andrew:] [LAUGHTER] Yeah. []
[John:] [cough] are a good team. So... those who think Liverpool... is a good one, standing in in that circle, we've got say... twelve thousand.... Okay. In the Everton circle we've got say... fourteen thousand.... And in the Tranmere circle we've got erm... twenty thousand say.... Okay. And we've got... thirty thirty thousand... people there... altogether. Now so I want to know the sort of the numbers in all the boxes. Now
[Andrew:] Right.
[John:] talk about it a bit before you start it. Erm is it similar to the problem you've done? Is it harder? Is it much harder?
[Andrew:] Well it's just erm... it's like... doing three of the previous ones isn't it like?
[John:] Yeah. How are you going to sort that out?
[Andrew:] Erm well I'll start off labelling these W... X Y Z.
[John:] Right....
[Andrew:] And er... label this. That's.......
[John:] So as you're labelling it, what do you think of... the problem?...
[Andrew:] Erm... it's er
[John:] Do you think it's easy or... very much harder than the last one or
[Andrew:] just er pretty difficult eh. [LAUGHTER] No....
[John:] You're not going to be able to just look at it and write down something equals two X are you.
[Andrew:] No.
[John:] You've got to really... work out what's what....
[Andrew:] Right.
[John:] Okay. So you've labelled your diagram and you this one is... that minus W minus Y minus X so on. Erm... it's a bit a bit awkward-looking isn't it?
[Andrew:] Yeah just a bit. [LAUGHTER]
[John:] Could you have labelled it in a way that would sort of help yourself more? Make it a bit simpler....
[Andrew:] those three added together, so W plus X plus... Y... plus... plus the er the rest of E... equal fourteen... thousand.
[John:] Yeah. Okay. So... what's your next step from there?
[Andrew:] Oh right, I write it down on the paper. So erm
[John:] Okay.... And how many equations are you going to get do you think?...
[Andrew:] Erm....
[John:] About.... About. Okay. All right.
[Andrew:] Hundreds. [LAUGHTER]
[John:] About hundreds. Erm
[Andrew:] No erm
[John:] Well okay let's look at it another way. How many equations would you l would you like?
[Andrew:] Mm about one. [LAUGHTER] Just one or two.
[John:] Erm... If I gave you a problem about sort of prices of apples and pears...
[Andrew:] I suppose it no I suppose erm two W... plus two three Y plus X... plus Z
[John:] Mm. If I said erm, Ten pears plus six apples costs two pound forty,
[Andrew:] Yeah.
[John:] how much does one apple cost? what would you say to that?
[Andrew:] You can't really do it really.
[John:] Why not?
[Andrew:] Cos you've got two unknowns.
[John:] And?
[Andrew:] And only one equation.
[John:] Right. So that give you any ideas about this.
[Andrew:] You got to have two two equations.
[John:] Two equations. How many unknowns have we got?
[Andrew:] Erm. Three four... five. No. That the I've got I I know these.
[John:] You know those.
[Andrew:] I've got four unknowns there.
[John:] So you've got four unknowns, so you're going to need
[Andrew:] Four equations right?...
[John:] Does that sound a bit daunting?
[Andrew:] Yeah. Well... so four simultaneous equations....
[John:] Mhm. And they won but they won't be difficult cos they're all linear.
[Andrew:] Yeah.
[John:] They're all there're no X squareds or W squareds or anything else in.
[Andrew:] .
[John:] But you can do them. So... with with that, you know how much the total comes to, thirty thousand. [whispering] Right. []
[Andrew:] So I could put Oh aye yeah, W plus e Y... plus X plus E... equals fourteen... thousand.
[John:] Mm....
[Andrew:] And erm....
[John:] Have you used Say in the first one,...
[Andrew:] Yeah. Er
[John:] you didn't use that. So can you use can you think of an equation that involves that?... The total people.... Well tell me tell from your diagram, what's the total?...
[Andrew:] Thirty thousand... that are there.
[John:] Right, but that okay?
[Andrew:] Oh just these added up?
[John:] Okay. So don't forget the one thousand who don't think any of them are a good team, add all those up and that will give you a fourth equation.
[Andrew:] Yeah.
[John:] And all you need to do then is to find one of these in terms of the others and substitute in, and it'll come out fairly easily. Now you can choose these... erm slightly differently. For example you've got lots of got lots of minuses in.... What you could have done say was... this one here that's marked E... doesn't have to be all of those who supported Everton, it could be... just that bit.
[Andrew:] Yeah.
[John:] Right. And you can an expression for it then in terms of that. But... there's no reason why you stop at three. It could have been twenty five different variables
[Andrew:] Yeah. [LAUGHTER]
[John:] there are in real problems. Erm... With two it's not too difficult. I mean sometimes you can almost just try... an intelligent guess and maybe
[Andrew:] Yeah.
[John:] about the third guess you try you get the right answer. With that, there's not much chance.
[Andrew:] No. [LAUGHTER]
[John:] Okay. Erm so you need to develop a system that's going to always give you a method and is always making the work easier for you and not getting you lost. Okay. So... that's enough I think...
[Andrew:] Mm.
[John:] on that... to... sort of show you the power of the system we were just using to erm... The obvious thing to do is to label every area of the diagram, and don't forget they do sometimes say, Seven were not interested in
[Andrew:] Yeah.
[John:] in anything. Label every area and then use what they give you about the total number. The total was thirty pupils,... add all that lot up equals thirty. And... when you were doing these... with lots of... you've got now got W X Y and Z in,... when you're adding up... if you lay them out like that
[Andrew:] Right yeah.
[John:] with the Ws under the Ws... it's you'll find that those horrible looking equations will simplify quite nicely cos there'll be a plus W and a minus W.
[Andrew:] ... I see yeah.
[John:] And you'll get things that... you know you'll get an equation just in X then.
[Andrew:] So it'll be erm... so fourteen thousand... minus W minus Y minus X
[John:] Mhm.
[Andrew:] and the next one then twelve thousand... minus W
[John:] Mm.
[Andrew:] minus Y, then...
[John:] Yeah.
[Andrew:] miss that gap plus Z
[John:] That's it. Exactly.
[Andrew:] And then that one
[John:] Right, you've got it.
[Andrew:] two thousand plus
[John:] Yeah.
[Andrew:] Y... and X... and Z.
[John:] So although it looks like, Ooh this is horrible, four equations four unknowns, I'll be here all night
[Andrew:] And then
[John:] it's not cos you're just.
[Andrew:] I've got
[John:] You're eliminating one at a time.
[Andrew:] Yeah.
[John:] Once you've got that fourth equation in from all this lot adds up to the total, and that's the one... that you seem to be forgetting, that's the one you've got to try and... remember. Okay?... So shouldn't be any problem on those.
[Andrew:] This one er I wasn't too sure either.
[John:] Mm looks like geometry to me.
[Andrew:] Mm yeah yeah.
[John:] So [reading] A B is parallel... to D C. Calculate the values of erm write down possible [] Okay. Erm it's geometry. That's the answer to that,
[Andrew:] Yeah.
[John:] basically. It's something that I usually say, Leave till the last. It is a Did you spend much time on that?
[Andrew:] No I j just had a l er...
[John:] Good.
[Andrew:] er an attempt and then I thought, No I'm not
[John:] Good.
[Andrew:] gonna It's a waste of time.
[John:] It's it's it's the sort of thing that I would always recommend
[Andrew:] And there's only about three marks for it anyway so.
[John:] Exactly. Leave your attempt till the end. So rather than you going well on this nice question, and you could you could have done it and picked up
[Andrew:] Yeah.
[John:] another sort of six marks or maybe eight marks for the the end bit of the question, erm you'd spent a bit of time on this. Anything to do with geometry... unless you've had a lot of practice and you're very good at it, you can finish up wasting time.
[Andrew:] Yeah.
[John:] So I would say, Have a go but leave it till last, when you when you've done everything else and maybe you when you've just checked through to see if the others are okay. So let's have a a little look here. [reading] Political party making election promises. [] [breath]... [LAUGHTER] Erm was that one alright?
[Andrew:] Yeah. That wasn't too bad.
[John:] [reading] Eight percent every year. []...
[Andrew:] a hundred and seventy five billion.
[John:] Mhm.
[Andrew:] Erm... put it over a hundred times the eight.
[John:] Okay.
[Andrew:] And then you add that onto that. And the next
[John:] Right.
[Andrew:] year you get
[John:] Now. Okay. Erm... Mm, yeah that's good. And did you do it that way, you worked out the percentage and added it on?
[Andrew:] Yeah. Added it on after
[John:] Right. What's... Start off with a hundred pound
[Andrew:] Yeah.
[John:] ... and add twenty percent on.
[Andrew:] Yeah.
[John:] What's the final amount?
[Andrew:] Hundred and twenty.
[John:] Okay. Start off with a hundred pounds and add... thirty five percent on.
[Andrew:] So a hundred and thirty five pounds yeah.
[John:] Okay. Start off with erm... two hundred and add thirty five percent on. What's your final answer?
[Andrew:] Erm... two... Two hundred
[John:] Okay.
[Andrew:] and seventy.
[John:] So how are you working it out?
[Andrew:] It's erm two hundred over a hundred
[John:] Two hundred and then work out... thirty five over a hundred... of that,
[Andrew:] Er
[John:] work out what that comes to, which is seventy, and then add your two hundred on. Okay. Well if we do something like... what we finish up with is... thirty five percent of two hundred... [whispering] of two hundred [] okay, plus a hundred percent isn't it?
[Andrew:] Yeah.
[John:] Of two hundred. So what does that come to?... A hundred and thirty five percent.
[Andrew:] Oh I see yeah.
[John:] Okay.
[Andrew:] .
[John:] So if you want to So it's if you when especially as you're going to do it on your calculator anyway,
[Andrew:] .
[John:] what would you multiply to get the answer straight off? What would you multiply that two hundred by?
[Andrew:] One three five and then percentage.
[John:] One point...
[Andrew:] Three oh three five.
[John:] three five. Yeah, or a hundred and... thirty five percent as you said. Yeah. Erm if you... find you've got a... calculator without a percentage key, and some of them haven't, especially
[Andrew:] Yeah.
[John:] some of the scientific ones,
[Andrew:] mine's got one.
[John:] erm you can just do that.
[Andrew:] .
[John:] So if I wanted to find out erm... Let's do that one with the the N H S, and this time I want to know how much it's going to be It starts off at a hundred pounds.
[Andrew:] Yeah.
[John:] A hundred billion. Okay, just call it a hundred. And we increase it by ten perc Well let's say we increase it by It's not out money is it, so let's really spend it and increase it by fifty percent every year.
[Andrew:] Yeah....
[John:] What would you multiply that by, that hundred?
[Andrew:] One point five.
[John:] Okay. So do that try that on your calculator.... So what did you get?
[Andrew:] hundred and fifty pound.
[John:] Right. So while that's still in, times one point five again.
[Andrew:] .
[John:] Well you've still got your one fifty in, so times one point five....
[Andrew:] So two hundred and twenty five right.
[John:] And leaving that in, times one point five.
[Andrew:] That's three hundred and thirty seven point five....
[John:] So you can see how they as if they ask you to do it over three years or five years or something
[Andrew:] You just keep on doing that yeah.
[John:] you don't you're not doing several operations and putting some in memory and bringing it back and giving yourself lots of chances of making mistakes.
[Andrew:] No.
[John:] It's just multiply it by that constant thing each time. And with most... with a lot of calculators I mean you Don't try it unless you really know your calculator and know how it works, you can just keep pressing equals,
[Andrew:] Yeah.
[John:] and it does the last thing you've done times the
[Andrew:] Yeah.
[John:] ... say do it once and say we want six years, put equals equals equals
[Andrew:] Yeah.
[John:] more times. Erm... if you're going to use thing something like that in the exam I mean there's no reason why you shouldn't use that multiplying it by one point whatever the percentage is.
[Andrew:] Yeah.
[John:] What would you have multiplied it by if it was erm the eight percent?
[Andrew:] One point eight.
[John:] One point... Well what would you
[Andrew:] Oh ze oh zero eight, yeah sorry zero eight.
[John:] Yeah, okay. That's the only thing to watch, that.
[Andrew:] Yeah cos that'd be
[John:] Eigh eighty percent...
[Andrew:] that'd be eighty percent yeah.
[John:] eighty percent is one point eight but eight percent, okay seven percent one point O seven. That's the only snag with that, that you you can get the wrong But it's very much way of doing it.
[Andrew:] Yeah.
[John:] ... Okay. Solve the equation.
[Andrew:] That's pretty straightforward that.
[John:] Expand that. You can do that.
[Andrew:] Yeah....
[John:] [reading] so much for its call-out... fixed charge plus the time. [] That's okay.
[Andrew:] Yeah.
[John:] Yeah?... And then working backwards. It costs
[Andrew:] Yeah.
[John:] sixty five pound
[Andrew:] And then just er just changing the subject there like.
[John:] Right. Now if we put that on there.
[Andrew:] Erm... I dunno this is somebody else's paper so... This is before I had a I worked it out as... I I th I thought it'd work out a three- four-five triangle.
[John:] Mm. Did you work out it was a ninety degrees?
[Andrew:] Erm... yeah well I I
[John:] Why?
[Andrew:] I realized that cos
[John:] Why? Cos it looked it?
[Andrew:] No cos erm... the here you've got this in a semicircle,
[John:] Right.
[Andrew:] and if it's touching
[John:] Okay.
[Andrew:] the top it's always ninety degrees.
[John:] Did you say that on the paper?
[Andrew:] Erm... I just put a little arrow with ninety degrees in the semicircle.
[John:] Mm. Cos it looked it.... Erm wh what they're looking for in this answer is erm... Because it's the angle in the semicircle... angle B is ninety degrees.
[Andrew:] Yeah.
[John:] And then carry on and do the other bit. Erm... you know you can look at that or you can put a protractor on it and
[Andrew:] Mm.
[John:] think Well that's ninety, but you need to say that it is ninety because it's the angle in there.
[Andrew:] Yeah.
[John:] Okay.... Right. So you'd probably get... minimum marks for that.
[Andrew:] three marks for that anyway.
[John:] Mhm....... What did you think of this?...
[Andrew:] I looked at it and I er panicked. [LAUGHTER]
[John:] [LAUGHTER]
[Andrew:] Then I looked at it again and I realized that it wasn't too hard.
[John:] Mm.
[Andrew:] I mean I got er the only th part... erm... I did mess up a bit was the... the part here.
[John:] Mm.
[Andrew:] Cos I got this the wrong way round. But other than that I got the... right.
[John:] Right. Now this... this... what I'd like you to do,... is read it out... but... only read the absolute bare bones of it. We don't want to hear anything about... Janet having trouble with her bike and [reading] Janet a powerful young woman [] or [reading] riding the race of her life [], or any don't want
[Andrew:] Yeah.
[John:] any of that. It's just rubbish. When you when you're reading this through, you're having a quick skim through, you can can you can cross out, you can sort of totally obliterate
[Andrew:] Yeah.
[John:] what you don't want to know because it's One of the problems here is there's so much noise.
[Andrew:] Yeah.
[John:] There's so much random stuff that you don't just details irrelevant.
[Andrew:] yeah.
[John:] You don't want to know that she was wearing odd socks and one of them was green, it's
[Andrew:] Mhm.
[John:] nothing to do with drawing the graph of... distance against time. So what are the important points in that?
[Andrew:] That they set off neck and neck.
[John:] Okay set off neck and neck. Right.
[Andrew:] Alright. [reading] Then Janet took the lead then overtaken ten seconds later. []
[John:] Mhm.
[Andrew:] [reading] Then built up a lead of fifty metres. []
[John:] Right....
[Andrew:] And so she... [reading] behind all the way but caught up with her a few metres []...
[John:] [reading] Before the end. []
[Andrew:] [reading] before the end. [] Oh [reading] just before the finish, fifty metres from the finishing line. []... And then [reading] Janet overtook her just before the finishing line. []
[John:] Right. So all this about the North Rose Trophy and... it's just... When you see You f you said you felt like panicking.
[Andrew:] Yeah.
[John:] You're in middle of an exam, you're trying to do everything very quickly, and there's a great long screed of text...
[Andrew:] Mm.
[John:] to plough your way through. Erm so... get sort of practise reading it so you can cut out the irrelevant stuff. And a good thing as I say is to just... erm highlight the stuff that you want if you've
[Andrew:] Yeah.
[John:] got a highlighter, or maybe underline it, and maybe even cross out.
[Andrew:] Yeah.
[John:] Because you rea you're going to read this about three or four times... aren't you?
[Andrew:] Yeah.
[John:] before you understand what on earth is going on here. [reading] They started off neck and neck. [] They started off together. Okay. This one goes in the lead and then stops and then the other one goes in the lead and then she puts a spurt on and catches up. You'll read that sort of round and round a few times before you get it straight in your head what's
[Andrew:] Yeah.
[John:] going on, and you don't need all this garbage in it as well. So your first time through you can cut some of that stuff out, and then you've got it. You've got the bones of the problem. Then you can work with it. Erm... so do you think you got it sorted out roughly? Even I mean... you didn't quite get it sorted out in the exam but
[Andrew:] No.
[John:] did you think that... you know what you've done wrong now,
[Andrew:] Yeah.
[John:] and how to do it? Yeah. There's there is often something on... distance time graphs and... also quite often it's like that. Now I think that's not a maths question.
[Andrew:] No. No.
[John:] That's an English question.
[Andrew:] Yeah.
[John:] I mean if you I don't know if you know of anyone whose standard of English is quite poor
[Andrew:] No.
[John:] but their maths are okay.
[Andrew:] Mm.
[John:] Now that's quite an unfair question.... Cos if they can't read through a lot of text and sort it out they haven't got much chance.
[Andrew:] No.
[John:] They just sit there mesmerized for half an hour and then
[Andrew:] [LAUGHTER]
[John:] Right time's up.... [whispering] Okay. []... So you're alright on trine... time train...
[Andrew:] Yeah.
[John:] [tut] train timetables. Mm?
[Andrew:] You just draw a... line of best fit... you know line of best fit through erm all the points well
[John:] Okay.
[Andrew:] the one that goes nearest the majority of points.
[John:] One goes nearest the majority of points. Okay.
[Andrew:] So if you've got a load of points all here
[John:] Mm.
[Andrew:] and there's one stray off here, you'd go through that way, the line of best fit.
[John:] Ignore any strays.... Mm.... Okay. How about the gradient of the line?
[Andrew:] That's like the... average... what is it it's a... Distance so it's the average... speed that's been travelled... throughout the journeys.
[John:] Mm. Erm ... the gradient of a line okay, there's a there's a bit of graph paper.... Okay X is... and there's a line.... What's its gradient?...
[Andrew:] Erm the change in the Y
[John:] Okay it
[Andrew:] so change in the...
[John:] It it's so measure measure the gradient of that... and tell me what it is....
[Andrew:] I'll make it so they're nice and e even....
[John:] Now you said you'd make it even.
[Andrew:] Yeah.
[John:] What did you mean by that?
[Andrew:] Oh... t I had two point nine by... three
[John:] Mm.
[Andrew:] by seven point nine.
[John:] Okay. The one you made nice and easy was the nice and even was the Y.
[Andrew:] Yeah.
[John:] Which one are you going to divide by?
[Andrew:] The Y, so change in Y over change in X so...
[John:] Mm. You're going to divide by the X.
[Andrew:] Oh yeah, best off doing the Yeah.
[John:] It doesn't matter cos you can....
[Andrew:] .... Better get rid of that line yeah. Probably pencil.
[John:] Okay. And what does that come to roughly?
[Andrew:] It's about erm... two point eight.
[John:] Right. Two point eight
[Andrew:] Seven centimetre.... So that's erm...
[John:] ....
[Andrew:] ....
[John:] What's what's the units?
[Andrew:] I don't know it's just whatever the...
[John:] Well what have you done? What did you do?
[Andrew:] Oh I cha I divided... Y over
[John:] You divided
[Andrew:] Divided the change in Y
[John:] So many... millimetres
[Andrew:] Yeah.
[John:] by another number of millimetres
[Andrew:] Oh right, oh millimetres that's it yeah.
[John:] and what's the answer?
[Andrew:] Erm millimetres.
[John:] Millimetres divided by millimetres is what?
[Andrew:] Yeah it's millimetres square. Millimetres... erm the other one.... What is it?
[John:] Erm... What's X divided by X?...
[Andrew:] One. Er one X. Just X.
[John:] Just one.
[Andrew:] It's just millimetres.
[John:] No it's just one. It's not millimetres, it's nothing. It's just a number.
[Andrew:] So that's just millimetres then yeah.
[John:] It's millimetres divided by milli No, it's not millimetres.
[Andrew:] No this.
[John:] The answer is not in millimetres. The answer is just a number.
[Andrew:] Oh right.
[John:] It's a ratio... of one length to another.... Okay?
[Andrew:] Yeah.
[John:] Find the ratio of... that height to that length.
[Andrew:] Mm.
[John:] Okay. And let's say it's a third.
[Andrew:] Mm.
[John:] It's not a third of metre, a third of a millimetre or a third of a kilometre, it's just a third, the ratio of that length to that one.... Okay?
[Andrew:] I see.
[John:] So the ratio of that length to that one is point four. But the gradient... the gradient of the hill... erm you could express it in terms of an angle couldn't you?
[Andrew:] Yeah.
[John:] If I say, Is that hill very steep? Ooh yes, about forty degrees.... So if we look at this...
[Andrew:] Twenty five.
[John:] Twenty five. Find the tan of twenty five degrees.... It won't be in centimetres or millimetres, it'll just be a number.
[Andrew:] Two point four six.
[John:] Right.
[Andrew:] Six three.
[John:] That's a much more accurate way of finding the gradient.
[Andrew:] .
[John:] Just find the tan of the angle.... You don't have to measure.... You've got two measurements here. This two point eight isn't accurate.
[Andrew:] No.
[John:] It's what, plus or minus... point one almost.
[Andrew:] Yeah.
[John:] I mean point one of a millimetre's not much. You could be almost well you could at least plus or minus point O five.
[Andrew:] Yeah.
[John:] Right. Which is a big percentage that you're And both of these could be out. One could
[Andrew:] Yeah.
[John:] be too big and one could be too small, which would make a big difference. But you can measure the angle pretty accurately on that to s to within say half a degree and have its tan.
[Andrew:] Right yeah.
[John:] Erm because... there's your angle,... there's that, and this is opposite over adjacent. So the gradient it's a ratio it's not... no units to it, not metres millimetres or anything else, and it's the tan of the angle. So the gradient... gives you a mea If I gave you the gradient, if I said the gradient is one, what would the angle be?
[Andrew:] Erm...
[John:] How would you find the angle?
[Andrew:] Just one and then... do the tan backwards.
[John:] Okay tan to the minus one.
[Andrew:] five.
[John:] So there's a that's that's what gradient means, the tie-up between them. It's not In some cases like erm the diagram you were doing about the... two girls running,... erm then the distance against time gradient will give you speed, give you velocity. We've done the Roses,... you're okay on the symmetry, you know about that now.
[Andrew:] Yeah.
[John:] I mean if if it hadn't got a,... there's noise in the problem there's
[Andrew:] Yeah.
[John:] irrelevant stuff. If you if you you strip it down to the
[Andrew:] Actually I didn't know how to do the arc. I wasn't sure how to do that.
[John:] Okay. But you know now. Erm...
[Andrew:] That was pretty st that the height of just a bit of
[John:] It's a bit of algy and a bit of pythagoras
[Andrew:] Yeah.
[John:] ... in together.... And the area.... What is how about that one?... What is the square root of six point four by ten to the... ten to the five?
[Andrew:] Well erm... eight hundred but when I did it... wrote it into normal terms like six point sixty four with one two three four noughts
[John:] Mm.
[Andrew:] and then I square rooted it.
[John:] Mm.
[Andrew:] Erm... yeah, I square rooted it.
[John:] Okay. Yeah that's a good way to do it. Mm slightly more easier for you is... just multiply it by one of those tens... so you've got sixty four by ten to the four. The square root of sixty then take the square root of
[Andrew:] Eight.
[John:] The square root of the sixty four is eight, the square root of the ten to the four... ten squared.
[Andrew:] Oh right yeah.
[John:] Yeah. Erm okay. It's a good way to do it to if you think, Oh I can't really handle this going on here, put all your noughts on and then you can... work out what they come to. Okay. Erm... weather stations near the north pole. we should start ringing alarms bells a little bit.... So what did you do for this one?
[Andrew:] Right I measured... well it's seven kilometres apart so I measured the distance between them.
[John:] Ah. Mm?
[Andrew:] Erm in centimetres or millimetres.
[John:] And it came to five. It wasn't seven [LAUGHTER] centimetres eh []?
[Andrew:] No so it's...
[John:] And it should have been shouldn't it?
[Andrew:] fifty millimetres.
[John:] Right. Okay, so you worked out what the scale was.
[Andrew:] [LAUGHTER] Yeah. []
[John:] And then what did you do?
[Andrew:] Erm... I... so it was... for one So seven kilometres... is equal to fifty millimetres.
[John:] Mhm.
[Andrew:] I did erm
[John:] Yeah. So you
[Andrew:] seven
[John:] work out your scale that comes to
[Andrew:] So seven over... five.
[John:] And then what did you do for this next bit?... It knows it's between... four and three kilom kilometres away.
[Andrew:] Yeah. So er point seven one er is equal to one kilometre.
[John:] Mhm....
[Andrew:] So work out the...
[John:] Yeah.
[Andrew:] So four kilometres so multiply point seven one by four.
[John:] Yeah. You worked out what that would be
[Andrew:] and then
[John:] on the one the scale and then what did you do?
[Andrew:] Yeah I worked out what this would be on the scale. that's from T so I dr drew a line... a cir a circumference.
[John:] A circle. Right.
[Andrew:] So I did round there.
[John:] Yeah.
[Andrew:] and a smaller one round there.
[John:] Right. So you work out
[Andrew:] .
[John:] what the scale is so you can set your... radius and draw your two circles and say, It's in there somewhere.
[Andrew:] Yeah.
[John:] Okay. That should get full marks.
[Andrew:] Yeah.... Bit.
[John:] Mm.
[Andrew:] Erm er that... trig trigonometry there again.
[John:] Mhm.
[Andrew:] And then er just work out the length of this and then the length of that and then subtract these two lengths.
[John:] So that's quite a nice question. Erm...... Seven marks on that. Okay.... Transformation.... You alright on the transformation?
[Andrew:] Yeah.
[John:] Yeah. You've been doing those alright haven't you.
[Andrew:] .
[John:] Ah. Mm.... How many of those can you get out of one of those eh?
[Andrew:] Erm
[John:] How many mugs of water to fill the tank?
[speaker001:] .
[John:] Okay. So you cal calculate the volume.
[Andrew:] Yeah.
[John:] That one's no problem.
[Andrew:] Yeah.
[John:] You think, Right I've done that with me fish tank.... That one. What did you do for that?
[Andrew:] Erm... I used the equation sheet which he wri wrote on the board.
[John:] Right. Okay which would have been on the front of your paper.
[Andrew:] Yeah but he we didn't have any
[John:] Right so as soon as you see that you think, Ah volume of a cylinder, I know, whee! straight to the front.
[Andrew:] Look at it and then erm... measured it... I did I d and then I redid this one and put it in the same scale,
[John:] Yeah.
[Andrew:] ... and then erm the answer I got for this
[John:] Mm. So
[Andrew:] divided it by that.
[John:] you what did you you used centimetres here did you?
[Andrew:] Yeah.
[John:] Right. Good. Because that thing you did with the fish tank has shown
[Andrew:] Yeah.
[John:] you that you can't say, Well one centimetre is a hundredth of a metre so
[Andrew:] No.
[John:] ... one cubic centimetre's going to be a hundredth of a cubic metre.
[Andrew:] [LAUGHTER]
[John:] it's obvious. [LAUGHTER] It's not it might be obvious but it's not true. Okay that that's the big thing they're looking for there. Convert it to the same units, use the formula and then... I mean that's a gift that isn't it?
[Andrew:] Yeah.
[John:] Mm? Six marks there for nothing really.... [LAUGHTER] Imagine them all bringing their mugs to fill the tank.
[Andrew:] Mm....
[John:] Now you've got to rework it again in cubic metres
[Andrew:] Yeah.
[John:] and work out how many seconds.... There's nine marks on that
[Andrew:] Yeah.
[John:] really a gift wasn't it?
[Andrew:] Yeah
[John:] I mean it's a it would be a joy to do it as well cos you think, I know how to do these.
[Andrew:] Yeah isn't this easy, this is relaxing. [LAUGHTER]
[John:] Okay. Erm I think you know there how to pick up quite a few...
[Andrew:] Quite a few more marks yeah.
[John:] quite a few marks without you learning anything
[Andrew:] No.
[John:] that you don't know already.... Yeah?
[Andrew:] Yeah
[John:] It's just sort of looking at 'em a bit. Erm the geometry one that we didn't look at,... you might be able to spot it in minutes,
[Andrew:] Yeah.
[John:] you know seconds even. You might be still on it twenty minutes later.
[Andrew:] Yeah.
[John:] So keep it for the end and keep it for when you've not only done all the questions but had a quick check through to see have
[Andrew:] [whispering] Yeah. []
[John:] you done something daft.
[Andrew:] Yeah.
[John:] Cos that's where the marks are thrown away. Not
[Andrew:] Yeah.
[John:] not something you can't do but something you've done ten times before.
[Andrew:] You just make a one silly mistake like. [LAUGHTER]
[John:] And you just think you know you like you've put metres instead of centimetres on the
[Andrew:] [LAUGHTER]
[John:] the hosepipe or something like that and you think, Oh that's not right, that you will spot. Like when you write a letter and you read it through and you think, Ooh... I've missed a word out here,
[Andrew:] Mm. [LAUGHTER]
[John:] or I've you know run two words in together or something. So that few minutes at the end is important for picking up these extra marks you've been trying to throw away, [LAUGHTER] okay, where you can do it, and the few minutes at the beginning is to look through and find that question... wherever it was, that one... on the last page.... Lot of marks going for it, dead ea And again it's a s it's a joy to do this sort isn't it.
[Andrew:] Yeah.
[John:] Cos you can see where you're going all the way through,
[Andrew:] Yeah.
[John:] you know that you know it it's nice
[Andrew:] It's just like nice and easy.
[John:] and... you feel really confident about it, you think, Wow this is piling up the marks.
[Andrew:] [LAUGHTER]
[John:] So... look for the good ones... don't spend too much t There'll only be one little geometry thing
[Andrew:] Yeah.
[John:] fiddling about with triangles and circles and stuff like that. So have a go
[Andrew:] And it's only about two or three five marks.
[John:] Exactly. So have a go at it when you've done the others.... Okay. Well I'd better get off. ... another one to get to soon. That's mine. That's yours.
[Andrew:] Right.
[John:] Practise with that protractor,
[Andrew:] Yeah.
[John:] so that you get used to it so that you prefer to use it to the other one. Have have have both, but if you've got bearings...
[Andrew:] It easy to b better.
[John:] use that. If someone gives you a... polygon to draw
[Andrew:] Yeah. Use that cos it's not not fiddling
[John:] Yeah.
[Andrew:] around twisting it.
[John:] I mean in the time is takes you to do a sketch you can draw accurately
[Andrew:] Be easier to.
[John:] draw round it, [taps table] dot dot dot,... draw even if you then sketch it without using a straight line, if you just join up the dots freehand,
[Andrew:] Yeah.
[John:] you get a very good sketch and you can see what's going on and you can see the angles.
[Andrew:] .
[John:] That's your one. Erm... Next week. What's happening about that? Where are we?
[Andrew:] I'm... on holiday then so er
[John:] You're on holiday. Right.
[Andrew:] Yeah.
[John:] Okay. So is that for What about the week after that?
[Andrew:] Erm... no I think I've got about two weeks off.
[John:] Okay, so I'll see you... erm
[Andrew:] So we break up on Wednesday that's... Friday Friday, So three Fridays.
[John:] Three Fridays.
[Andrew:] Yeah..
[John:] So I'll see you in about... I'd better work out what the date is. Erm
[Andrew:] Can have a look on the calendar in there.
[John:] Okay. And probably be an idea if you remind me sometime during that week when I'm due to see you.
[Andrew:] Alright yeah I'll give you yeah.
[John:] Just give me a just give me a ring. But er we'll we'll skip the next three then.
[Andrew:] Alright yeah.
[John:] Okay. I better remember my... tape recorder.
[Andrew:] Okay. Erm I've found out when the exam is.
[John:] Yeah?
[Andrew:] It's erm... right at the end of the... month of the rest of the exams....
[John:] Okay.
[Andrew:] So erm... Yeah it's well well at the end of the month.
[John:] So have you got what's what's the best... [tape ends]