[speaker001:] .
[Malcolm:] Right.
[speaker003:] That erm... course work's gotta be in a week today.
[Malcolm:] Yes. What've you Is er
[speaker003:] I've got that other one to do yeah.
[Malcolm:] What both sets of course work have gotta be in?
[speaker003:] Mm.
[Malcolm:] Right in that case we'd better have a look at that now....
[speaker003:] Is it in here? No it's in the other one....
[Malcolm:] Have you got the other one written up?
[speaker003:] Yeah it's just that erm... It's just got.
[Malcolm:] Right. Yes.
[speaker003:] And that's all.
[Malcolm:] Lovely....
[speaker003:] .
[Malcolm:] [whispering] Next one. []... should have plenty of work there, you should be okay
[speaker003:] Mm.
[Malcolm:] with that.
[speaker003:] that one. [cough]
[Malcolm:] Which one are we going to do? Can't remember....
[speaker003:] [cough] [cough] volumes or polygons....
[Malcolm:] [reading] Cheapo supermarkets produced dried peas in sealed polythene bags contained in er stored in wire bins blah blah []... Oh. [sigh]... Was this one we were wanting to look at wasn't it?
[speaker003:] Yeah....
[Malcolm:] [reading] Draw accurately and name six polygons with sides three four five six seven and eight and nine sides. Three of these should be regular. []...
[speaker003:] Mhm.
[Malcolm:] Okay. So y can you remember how to erm produce er polygons, how to how to make them? How to make regular polygons?
[speaker003:] No.
[Malcolm:] Right. Now a regular polygon... Where are we? Something to write with.... A regular polygon... has all its sides... and... all its angles... equal....
[speaker003:] Mhm.
[Malcolm:] So if you start with three,... that's
[speaker003:] Mm.
[Malcolm:] a triangle isn't it? It's the smallest polygon you can make cos if you've got two sides you can't join it up can you?
[speaker003:] No.
[Malcolm:] So you've three sides,... all equal length, and all the angles have gotta be equal. Now the triangle add ups to how many degrees?
[speaker003:] ... Hundred and eighty.
[Malcolm:] Hundred and eighty. So if the three have gotta be equal you've a hundred and eighty divided by three which is sixty each.
[speaker003:] Mhm....
[Malcolm:] Er and then the next one's four. Now there are a lot of four-sided ones really,...
[speaker003:] Mm.
[Malcolm:] erm cos if you start thinking about the four-sided ones there's the square which everybody... knows about which is again is the regular polygon isn't it?
[speaker003:] Mm.
[Malcolm:] Cos it's got all four sides equal and four ninety degrees.
[speaker003:] Mhm.
[Malcolm:] Then if you can think about that as being stretched as it were,
[speaker003:] Mm.
[Malcolm:] you know keeping two of them fixed and the other two If you think about it being stretched you get a rectangle don't you?
[speaker003:] Mhm. [cough]...
[Malcolm:] A K A an oblong. But again
[speaker003:] Mm.
[Malcolm:] ... it's not regular.
[speaker003:] no.
[Malcolm:] But you can then think of that That's with the... the sides... two sides stretched but still parallel to one another aren't they?
[speaker003:] Yeah.
[Malcolm:] Now you c the angle's ninety. As soon as you destroy the ninety degree angle you get a parallelogram don't you?...
[speaker003:] Yeah. [LAUGHTER]
[Malcolm:] [LAUGHTER] I'm gonna have to draw it there cos I wanted to... Like that. You get the parallelogram.
[speaker003:] Yep.
[Malcolm:] Opposite sides parallel, but the angles aren't the same any more.
[speaker003:] No.
[Malcolm:] Now if you think about going this way, that was... So keeping the angles constant at ninety wasn't it?
[speaker003:] Mm.
[Malcolm:] Now this way you can keep the sides constant and destroy the angles.
[speaker003:] Mm.
[Malcolm:] And if you do that you end up with a rhombus... which is a squashed
[speaker003:] Mm.
[Malcolm:] square. The four sides are the same length,
[speaker003:] Mm.
[Malcolm:] but the angles have gone. So that's a rhombus.... And that's your quadrilaterals except for this one of course.... haven't finished yet with quadrilaterals.... There's this one... which
[speaker003:] Yeah.
[Malcolm:] is the kite.
[speaker003:] Yeah....
[Malcolm:] It's got pairs of equal sides,...
[speaker003:] Mm.
[Malcolm:] but the two pairs of equal sides aren't opposite one another so you don't end up with the the either the rectangle
[speaker003:] Mm
[Malcolm:] or the parallelogram. If you think about this you can think of a kite as two isosceles triangles glued together.
[speaker003:] Mm.
[Malcolm:] Of different sizes. As you can think of a rhombus as two iso two same-sized isosceles
[speaker003:] Yeah.
[Malcolm:] triangles glued together. So it's line of symmetry there and that isn't a line of symmetry.... Okay? So there's your four....
[speaker003:] Yeah.
[Malcolm:] That's the regular one. You can get thr triangles isosceles triangles,... right-angled triangle and
[speaker003:] Mm
[Malcolm:] scalene and so on. There's your four... That's your quadrilaterals, except of course for this one. Any old size.
[speaker003:] Mm.
[Malcolm:] No no angles the same, no
[speaker003:] Mm.
[Malcolm:] no lengths the same, which is the quadrilateral. Which is the general one. [LAUGHTER]... All right?
[speaker003:] Mm.
[Malcolm:] Now it is possible... to draw these, and the the next one up is a pentagon isn't it? Can you remember? You've gotta remember these. You need to remember these anyway for your exam.
[speaker003:] Mm.
[Malcolm:] Well if we're gonna draw a pentagon... It's easy enough, we want five sides, so everybody can draw a house end....
[speaker003:] Mm.
[Malcolm:] But it's not regular.
[speaker003:] No.
[Malcolm:] You can also draw this, and people tend to forget this one. Oh by the way, sorry, there's another one here.... I tend to forget it, everybody else tends to forget it, it's an arrow, arrowhead, or something like that,... is also a quadrilateral. It's got four sides.
[speaker003:] Mm.
[Malcolm:] So you can remem you can think about this. This is can happen here as well.
[speaker003:] Mm.
[Malcolm:] So you can actually get... That's a pentagon... isn't it? Five sides.
[speaker003:] Mm.
[Malcolm:] And you can stretch it and bend it wherever you want but that's th th
[speaker003:] Mm.
[Malcolm:] it's it's till a pentagon isn't it? It is possible of course erm to you know draw a line er c you know distort it, like that.
[speaker003:] Mm.
[Malcolm:] But if you want to draw the regular one... You can sit and do this backwards and work out what the angles are.. But if you think, all polygons can be drawn inside a circle.
[speaker003:] Mm.
[Malcolm:] So if you draw a circle... You know you wanna split this up into five sides....
[speaker003:] Mm.
[Malcolm:] So, a circle is three hundred and sixty degrees isn't it?... So if you wanna split that up into five segments you divide three hundred and sixty by five which gives you? Seventy two. Yeah?
[speaker003:] Mm.
[Malcolm:] Seventy two degrees, you want five sides.... So if you start at the centre... and draw a line and mark off seventy two degrees,
[speaker003:] Mm.
[Malcolm:] and then mark off another seventy two degrees and another seventy two degrees and another seventy two degrees and join the ends,
[speaker003:] Yeah.
[Malcolm:] you'll end up with a regular pentagon.
[speaker003:] Sorry.
[Malcolm:] Have you got... a compass and protractor to hand?
[speaker003:] No. I can go and get one....
[Malcolm:] I don't need wonderful drawing instruments but [LAUGHTER]... [break in recording] No no no. No, draw yourself a circle.
[speaker003:] Oh.
[Malcolm:] Try one of those.... Right. Okay. Er put a line in from somewhere, there from the centre up to there, just to give yourself a start somewhere.
[speaker003:] Mhm.
[Malcolm:] Just draw that in.... Straight line, yeah, use the edge of the protractor'll do.
[speaker003:] Mhm....
[Malcolm:] Now draw yourself seventy two degrees from that. Seventy two degrees seventy two seventy... er all the way around.... That's right.... Join each end there, you end up with a pentagon. Just join join them up and see, what d'you get?... Alright?
[speaker003:] Mm.
[Malcolm:] You get a very nice... pentagon.
[speaker003:] Mm....
[Malcolm:] You can do the same thing for any of the others, so however man many sides you want,...
[speaker003:] Mm.
[Malcolm:] you divide three hundred and sixty by that number of sides which gives you this angle here,... the angle at the centre. Cos you just dividing three hund You just divide the three hundred and sixty up into seven eight or how many bits
[speaker003:] Mm.
[Malcolm:] it is. You've got a slight problem with things like seven cos what's three hundred and sixty divided by seven?
[speaker003:] I don't know.
[Malcolm:] Yes I don't know either! [LAUGHTER]
[speaker003:] Fifty four point fifty one point four.
[Malcolm:] Fifty one point four. Now if you're drawing fifty one point four,...
[speaker003:] Mm.
[Malcolm:] you're going to have to... erm you know very very careful with a protractor and the protractor you've got not it's not gonna really do a fifty one point
[speaker003:] Mm.
[Malcolm:] four, it's gonna be marginally out.
[speaker003:] mm.
[Malcolm:] Fifty one point five you'll f you'll manage...
[speaker003:] Mm.
[Malcolm:] reasonably, particularly if you got a sharp pencil. So... But six you can do. Can you do can you do a hexagon the other way, which is six sided?... S you could do... you could do a hexagon this way actually draw a hexagon. Do another circle.... Now,... it's gotta Hexagon's got six sides hasn't it?
[speaker003:] Mm.
[Malcolm:] So you've gotta divide three hundred and sixty by six. What d'you come to?...
[speaker003:] Sixty.
[Malcolm:] Sixty. So if you need sixty just draw it and I'll show you You'll remember the other way when I've sho when I've shown you it. Just draw that up. [break in recording] D'you remember drawing the flower pattern when you were a kid? [LAUGHTER]...
[speaker003:] No.
[Malcolm:] Eh? Oh dear me!... Can you remember?...
[speaker003:] Oh yeah.
[Malcolm:] Yes?... That's right, just mark them off. Now this is n not quite... accurate, because it doesn't quite go into it six times. It's just marginally off. But usually by the time you've taken in the errors of your compass and your pr your pencil and everything else it come out to be fairly close to. Do you remember doing that?
[speaker003:] Yeah.
[Malcolm:] That's right, well that's in that's it. That's all it is. And then join 'em up. But it only works for that particular...
[speaker003:] Mm.
[Malcolm:] erm... f... figure for the hexagon. It actually doe it works for triangle, you can do a triangle as well cos of course if you realize you can put a line across there a line across there a line
[speaker003:] Mm.
[Malcolm:] across there, which'll give you the equilateral triangle. So you understand what you've got to do. You've only gotta pick out... Does it say three regular ones?... [reading] Three of these should
[speaker003:] Mm.
[Malcolm:] be regular. []
[speaker003:] I've gotta draw six polygons, and then
[Malcolm:] Yes. Yes. Right. So the first polygon's a triangle. The next one's a quadrilat well is a quadrilateral or a square or whatever it is. Fifth one That one's a pentagon. That's a hexagon....
[speaker003:] Mm.
[Malcolm:] What's seven?...
[speaker003:] S... What do you call it?
[Malcolm:] Well that's the point. You've gotta know what you call it.
[speaker003:] [LAUGHTER]
[Malcolm:] I think it's a heptagon.
[speaker003:] .
[Malcolm:] I think so. There's also people who call it a septagon as well September seventh.
[speaker003:] Oh.
[Malcolm:] You know that. That sort of thing. Erm... usually called a heptagon. You're gonna have to look it up, you're gonna find a
[speaker003:] Mm.
[Malcolm:] erm... maths dictionary or something somewhere. Eight is the?...
[speaker003:] Erm... the octagon isn't it?
[Malcolm:] It's the octagon yes. And nine's a nonagon.
[speaker003:] Mm. [LAUGHTER]
[Malcolm:] Well nine's non. Yeah?
[speaker003:] Mm.
[Malcolm:] N O is actually the the the November is actually
[speaker003:] Mm.
[Malcolm:] really the ninth month except they... decided to stick a couple of m extra months in to get the twelve. It was originally September October November and December are seven eight nine ten.
[speaker003:] Mm....
[Malcolm:] Really. [LAUGHTER] Cos that's why they d You know it's it's sept
[speaker003:] Mm.
[Malcolm:] oct nov and dec, seven
[speaker003:] Mm.
[Malcolm:] eight nine ten. But
[speaker003:] Yeah.
[Malcolm:] they stuck a couple of months in the middle of the year
[speaker003:] [LAUGHTER]
[Malcolm:] so that th instead of working properly
[speaker003:] Mm.
[Malcolm:] [LAUGHTER] they're now two out. Right. [reading] polygon triangle as shown. Find first by measurement and then by calculation the sum of the internal angles. [] Er... right. If you got some... There's a pentagon.... It says divide 'em up.... Sum [reading] Find first by measurement and then by calculation []. Right. Okay. I'm gonna have to draw these with straight lines with a ruler it's not gonna work is it?... Can you draw one of these? Draw a pentagon with a with a ruler.... Doesn't matter what it looks like as long as it's got five sides that join together without gaps. [LAUGHTER]... So now you've gotta split it up into triangles by starting at one corner.
[speaker003:] Mm.
[Malcolm:] So pick a corner,
[speaker003:] Mhm. Mm.
[Malcolm:] and split em up, yeah? That'll do.... Draw a line so you produce triangles. You've already got one there haven't you?
[speaker003:] [LAUGHTER]
[Malcolm:] How many triangles have you got?
[speaker003:] Three.
[Malcolm:] You've got three haven't you?
[speaker003:] Yeah.
[Malcolm:] Five sides, three triangles.... It wants the sum of the internal angles. Now the internals angle is that isn't it?
[speaker003:] Mhm.
[Malcolm:] And that and that and that and that. So can you measure those,... write hem down and add them up. See how good you are at measuring ang how [LAUGHTER] how good that protractor is.
[speaker003:] [LAUGHTER]... That one's... sixty one isn't it.
[Malcolm:] Yeah sixty one.... Yes.... That one's not far off ninety I think....
[speaker003:] smack on ninety.
[Malcolm:] By jove! [LAUGHTER]
[speaker003:] [LAUGHTER]... Er that's
[Malcolm:] Well what's your Just Long as long as you stick That's it....
[speaker003:] One hundred and seven.
[Malcolm:] Yeah.... Next time I shall bring you a protractor that you can read.
[speaker003:] [LAUGHTER] I bought I had one but I think someone nicked it.
[Malcolm:] Yes. It tends to be the fate of protractors.
[speaker003:] Mm. Hundred and five.
[Malcolm:] Yes I'd call that a hun I'd go along with that at a hundred and five....
[speaker003:] Hundred and... nine.
[Malcolm:] Hundred and nine all right then. Just tot 'em up and see what you what you the total comes to....
[speaker003:] Four hundred and seventy two.
[Malcolm:] Four hundred and sev Eh?... You sure? [LAUGHTER]
[speaker003:] ....
[Malcolm:] That must be more than sixty three.... Sixty one. Measure that one again please....
[speaker003:] ....
[Malcolm:] That's right....
[speaker003:] a hundred and twenty. [LAUGHTER]
[Malcolm:] That's more like it yes. [LAUGHTER]... That's much more like it....
[speaker003:] Five hundred and thirty two.
[Malcolm:] Five hundred and thirty two. Okay. Now, it says by calculation. Now, what do you know about the angles of a triangle?
[speaker003:] Add up to a hundred and eighty degrees.
[Malcolm:] Add up to a hundred and eighty degrees. So you've got a hundred and eighty degrees there haven't you? You've got that one plus these two gives you a hundred and eighty. You've got a hundred and eighty degrees there haven't you?
[speaker003:] Mm.
[Malcolm:] Cos these... are these three add up to a hundred and eighty. And you've got a hundred and eighty degrees there. These three add up to a hundred and eighty. So all in all you've got ever angle covered here haven't you?
[speaker003:] Mm.
[Malcolm:] So it should come to three times one hundred and eighty.
[speaker003:] Mm.
[Malcolm:] Which is?...
[speaker003:] Five hundred and forty.
[Malcolm:] Five hundred and forty. You're eight degrees out, which is not bad considering this the [LAUGHTER] rather poor nature of your erm protractor [].
[speaker003:] [LAUGHTER].
[Malcolm:] So when you do this properly
[speaker003:] Mm.
[Malcolm:] make sure you've got a reasonable size reasonable clear protractor
[speaker003:] Mm.
[Malcolm:] and draw a reasonable sized diagram. Now...
[speaker003:] Mm.
[Malcolm:] Yeah. But of course the thing about it is that it works for any shape.
[speaker003:] Mm.
[Malcolm:] So no matter if you have... and pick a corner. I'd rather pick this one because it makes life easier,... you can see
[speaker003:] Mm.
[Malcolm:] because of this idea here. Go like that,... like that and like that, and you've got one two three four triangles. You should have six
[speaker003:] Mm.
[Malcolm:] sides, one two three four five six.
[speaker003:] Mm.
[Malcolm:] As long as you pick i If you pick your corner carefully,
[speaker003:] Mm.
[Malcolm:] you don't end up with problems. And if you draw your figure carefully as well if you did a erm
[speaker003:] Mm.
[Malcolm:] a hexagon like that with a... going inwards rather than sticking outwards,
[speaker003:] Mm.
[Malcolm:] you'll erm... you'll find you get nice easy...
[speaker003:] Mm.
[Malcolm:] erm triangles that come out of that. Now then.
[speaker003:] That's doing it by measurement and tha and that's doing it by calculation ain't it.
[Malcolm:] That's right, that's doing by measurement and that's doing by cal Y actually I think... dunno.... My eyes aren't as good as yours but me me glasses aren't bad. Oh I can't No I can't see this on this... protractor. No that's one of the problems.
[speaker003:] Yeah.
[Malcolm:] That's really one of the problems. Get a decent one.
[speaker003:] About ten P.
[Malcolm:] Well they're at least l you know... they're hardly the m greatest thing for... breaking the bank. So are you happy with what you're supposed to do with this to produce these results?
[speaker003:] Yeah.
[Malcolm:] Yes.
[speaker003:] that mea that measurement and back up again.
[Malcolm:] Yeah, yeah.
[speaker003:] Yeah.
[Malcolm:] [reading] Draw a graph... draw a graph by plotting the internal angle sum [], which is the sum of the internal angles, [LAUGHTER] in this case five hundred and forty,
[speaker003:] Mhm.
[Malcolm:] [reading] on the vertical axis [], upwards,
[speaker003:] Mm.
[Malcolm:] [reading] the number of sides, N on the horizontal axis. Use your graph to find the sum of internal angles of an eleven-sided blah blah and twelve sided [].
[speaker003:] Mm, so you just carry it on do you?
[Malcolm:] You just carry it on.
[speaker003:] Mm.
[Malcolm:] Just carry the graph on. [reading] Deduce a rule from your results so far that could be used to find the internal angle sum of a polygon with any number of sides. [] You should know now.... How many triangles?
[speaker003:] ... Three.
[Malcolm:] How many sides?
[speaker003:] Five....
[Malcolm:] I rest my case.... [reading] Using your original polygon constructions []...
[speaker003:] Mm.
[Malcolm:] [reading] that the sum... Show using your original polygon constructions that the sum of the external angles of a polygon is three hundred and sixty.... Mm.
[speaker003:] Mm.
[Malcolm:] Now you think about walking round this. I've
[speaker003:] Mm.
[Malcolm:] started along here, I turn through that, I walk along here I turn through that angle I walk along here I turn through that angle I walk along here I turn through that angle,... I walk along here and turn through that angle and I'm back where I started from.
[speaker003:] Mm.
[Malcolm:] Now in fact of course you've gone round in a?...
[speaker003:] Circle.
[Malcolm:] In a circle. A circle is... three hundred and sixty
[speaker003:] Mm.
[Malcolm:] degrees.
[speaker003:] Mm....
[Malcolm:] Which of course is what you were doing here.
[speaker003:] Mm.
[Malcolm:] Are you happy with that? Does that give
[speaker003:] Mm.
[Malcolm:] you a reasonable start?
[speaker003:] Mm..
[Malcolm:] Go on tell me
[speaker003:] mark this out in a table don't you?
[Malcolm:] Yes. So y y the internal angles of the regular ones... you can measure.
[speaker003:] Mm.
[Malcolm:] Don't when you're doing this This has six triangles in it hasn't it?
[speaker003:] Mm.
[Malcolm:] If you're doing it by calculation, you pick one corner and draw... tr
[speaker003:] Y
[Malcolm:] the other triangles. Don't do this, cos it doesn't work that way. You need to have picked one vertex and just draw
[speaker003:] Mm.
[Malcolm:] the diagonals from that vertex to the other ones.
[speaker003:] Yeah.
[Malcolm:] Otherwise you end up with a some very very strange results.
[speaker003:] Mm.
[Malcolm:] Right?
[speaker003:] Yep. That's that bit accomplished.
[Malcolm:] That's that bit accomplished yes.
[speaker003:] just got that one to finish off..
[Malcolm:] On the fourteenth.
[speaker003:] Yeah I've got seven actually. Seven or eight. I'll just go and get them. I've got them upstairs.
[Malcolm:] Right. [break in recording] Yeah..... [LAUGHTER]... That's right.... Yeah, but you're as tidy as I am you are!
[speaker003:] [LAUGHTER]
[Malcolm:] [LAUGHTER] Come on then.
[speaker003:] That's nought.
[Malcolm:] Nought, that's right.
[speaker003:] That's three.
[Malcolm:] Yes.
[speaker003:] Six.
[Malcolm:] Yep.
[speaker003:] That's... eight.
[Malcolm:] What's the sum of those three?
[speaker003:] Nine.
[Malcolm:] Nine that's better.
[speaker003:] And that's... eight... twelve.
[Malcolm:] Twelve is right....
[speaker003:] ... minus two.
[Malcolm:] Just write the minus two all the way across there cos it allows you to do the calculation much easier rather than trying to remember it. [LAUGHTER]
[speaker003:] Mm.
[Malcolm:] So nought minus two.
[speaker003:] Minus two.
[Malcolm:] Is Good. Well done.
[speaker003:] .
[Malcolm:] Eh?
[speaker003:] ?
[Malcolm:] Oh beg your pardon yes sorry. [LAUGHTER] You're right [LAUGHTER].
[speaker003:] ... Seven.
[Malcolm:] Mhm.
[speaker003:] .
[Malcolm:] Yeah. If you're gonna draw that graph up what will you have to do?
[speaker003:] put it down below.
[Malcolm:] You'd have to put it below. That's right.
[speaker003:] Mm.
[Malcolm:] Right now I don't wanna I'm gonna have to push you along a bit to... to erm make sure you get this done. So I don't... You know whilst I would have normally... if I'd been teaching a class done about five or six examples of that, that's all you're gonna get.
[speaker003:] [LAUGHTER] Right.
[Malcolm:] To start with, anyway. So... we're gonna up the ante a bit. We've got X Y equals X squared... plus two X, which is what I was talking about before.
[speaker003:] Mm.
[Malcolm:] We'll still do it the same. Nought one two three four.
[speaker003:] Mhm....
[Malcolm:] We're gonna set it up again. We got X... and nought one two three four, but now to start the calculation... we're going to put X squared in first.
[speaker003:] Mm.
[Malcolm:] So we've got X squared, and then we're gonna put in two X. That term then that term.... So, nought squared?
[speaker003:] It's nought.
[Malcolm:] It's nought. One squared? No.
[speaker003:] [LAUGHTER].
[Malcolm:] One times one.
[speaker003:] Oh yeah one.
[Malcolm:] One. Be very careful with that cos it's very very easy to do that. Yeah.
[speaker003:] Four.
[Malcolm:] Yep.
[speaker003:] Six. Nine.
[Malcolm:] [tut] How many?
[speaker003:] ... Nine.
[Malcolm:] Nine.
[speaker003:] Twelve. No it's not twelve.
[Malcolm:] Don't do that!
[speaker003:] [LAUGHTER] It's erm... sixteen.
[Malcolm:] It's sixteen. You must get over this business
[speaker003:] Mm.
[Malcolm:] ... of... thinking about it in those terms, that way along, along
[speaker003:] Mm.
[Malcolm:] that line. Think about it from there to there.
[speaker003:] Mm.
[Malcolm:] What's the relation that to that not this to this.
[speaker003:] Mm.
[Malcolm:] It's really very erm bad this business. You know you got introduced to it when you were... y very
[speaker003:] Mm.
[Malcolm:] small and the things you first hear are the things you remember.
[speaker003:] Mm.
[Malcolm:] [LAUGHTER] And it takes a terrible st struggle to get rid of it. Okay, so now we want two X....
[speaker003:] Two. Er two X? Oh that's nought.
[Malcolm:] It's nought, good.
[speaker003:] S Two. [LAUGHTER] Four.
[Malcolm:] Yes.
[speaker003:] Six.
[Malcolm:] Six.
[speaker003:] Eight.
[Malcolm:] Eight. And now we add 'em together.... Well I'll do the first one for you cos I can do that.
[speaker003:] [LAUGHTER].
[Malcolm:] [LAUGHTER].... Yes. What's the next one? That's a two.
[speaker003:] Three.
[Malcolm:] Three.
[speaker003:] Eight.
[Malcolm:] Eight.
[speaker003:] Fifteen.
[Malcolm:] Fifteen.
[speaker003:] Erm
[Malcolm:] Twenty four.
[speaker003:] Mm....
[Malcolm:] I'm the next one I'm gonna do I'm gonna spread it down below nought so we're getting into the negative numbers. But it w which will give you a better picture of this graph, but we gonna have to get... a chunk on this.... I'm also gonna open this up a bit. One... two... three... four. Just for reasons that it'll draw better if I open it.... I'm not gonna get it on........
[speaker001:] Yeah you will.
[Malcolm:] I'm not you know. Gotta get up to thirteen er up to twenty four.
[speaker003:] Oh right. [LAUGHTER]...
[Malcolm:] I'm not gonna go that far. Blah blah blah. We'll get enough from this. So nought is nought which is alright. One... two three four. One is gonna be three isn't it.
[speaker003:] Mm.
[Malcolm:] Which is there roughly. Two is going to be eight.... Which is about there isn't it?
[speaker003:] Mhm.
[Malcolm:] Three is gonna be fifteen.... Which is... One of these days I'll do this on graph paper. I'll try fifteen rather than fourteen. What you've got is this. It's not a straight line.... There's no way you can get a straight line through that
[speaker003:] No.
[Malcolm:] at all. So it's actually a curve... like that.
[speaker003:] Mm.
[Malcolm:] Yeah?... [whispering] other side the same. [] It's called a parabola. That's only one half of the parabola.
[speaker003:] Mm.
[Malcolm:] So what we'll do is to gi y Thing about this is you need the practice in in actually crunching
[speaker003:] Yeah
[Malcolm:] the numbers.
[speaker003:] Mm.
[Malcolm:] So what we'll do is we'll have... It's also actually equals X squared plus erm we'll make it three X... minus... No I'll not be I'll not be nasty.
[speaker003:] [LAUGHTER]
[Malcolm:] Plus one. We're gonna go from minus three to three.
[speaker003:] Mm.
[Malcolm:] Now again just in steps of one. Don't...
[speaker003:] Mm.
[Malcolm:] you know try and [LAUGHTER] get [] [LAUGHTER] Sometimes in an examination they'll give you funny numbers like two point five but they tend to work 'em out for you.
[speaker003:] Mm.
[Malcolm:] So you get things in there that have already been worked out for you so you don't have to... erm... So minus three. The next one will be minus two won't it?
[speaker003:] Yeah.
[Malcolm:] Next one?
[speaker003:] Minus one.
[Malcolm:] Minus one.
[speaker003:] Nothing.
[Malcolm:] Nought.
[speaker003:] One.
[Malcolm:] One...
[speaker003:] .
[Malcolm:] two and three.... So now we're gonna work out X squared. What's X squared then?... What's minus three times minus three?
[speaker003:] Six innit?
[Malcolm:] No! Three threes are? Nine.
[speaker003:] Nine.
[Malcolm:] Yeah so minus times a minus is plus.
[speaker003:] Mm.
[Malcolm:] That'll be nine.
[speaker003:] That'll be... four.... Two. And one.
[Malcolm:] One.
[speaker003:] [LAUGHTER]
[Malcolm:] [LAUGHTER] Well rethunk. Yes nought. [LAUGHTER]
[speaker003:] One.
[Malcolm:] One.
[speaker003:] Four.
[Malcolm:] Four.
[speaker003:] Nine.
[Malcolm:] Nine. Good. Now it gets hairy. But you may... Whatever,...
[speaker003:] Mm.
[Malcolm:] if you square a number you're gonna get a positive answer. It's gonna give you a plus every time. So if end up squaring a number and get a negative answer you've got it wrong!
[speaker003:] [LAUGHTER].
[Malcolm:] But now of course three times minus three is minus nine.
[speaker003:] Mm.
[Malcolm:] Three times minus two is?...
[speaker003:] Minus six.
[Malcolm:] Minus six. Minus three.
[speaker003:] Three.
[Malcolm:] Nothing,
[speaker003:] Mhm.
[Malcolm:] three, six
[speaker003:] Six.
[Malcolm:] and nine. Remember you're working from there to there.
[speaker003:] Mhm.
[Malcolm:] Now we want a plus one.... Nine minus nine plus one.
[speaker003:] Minus eight. No, add em all up?
[Malcolm:] Add em all up.
[speaker003:] Plus one.
[Malcolm:] Plus one is good.
[speaker003:] Sic... minus three.
[Malcolm:] Minus...
[speaker003:] Two. Not two.... Si
[Malcolm:] Four ones
[speaker003:] Four.
[Malcolm:] five, take away six is minus one.
[speaker003:] One yeah....
[Malcolm:] Minus one again.
[speaker003:] Minus one.... One.
[Malcolm:] One.... Five.... Eleven.... Nineteen.
[speaker003:] Mhm.
[Malcolm:] You with that?
[speaker003:] Yeah.
[Malcolm:] Happy?
[speaker003:] Yeah.
[Malcolm:] I've not gone too fast?
[speaker003:] No.
[Malcolm:] Good. [whispering] That's okay. [] So now of course we want minus three to plus three, so we want the axis down the middle....
[speaker003:] Mm.
[Malcolm:] Don't we? And also we've gotta go down to minus one,... so we'll put it there and hope and pray we've got enough on this time.... Nought, one, two, three, minus one, minus two, minus three. [sigh] Oh. [sigh] Yeah. We'll put in minus one there. [watch beeps] One, two, God is it that time? three, four, five, six, seven, eight, nine, ten, eleven, twelve. It's getting better.
[speaker003:] [LAUGHTER]....
[Malcolm:] Done it! [LAUGHTER]
[speaker003:] [LAUGHTER]
[Malcolm:] Got it on.... Right. Minus three is at one isn't it?
[speaker003:] Mhm.
[Malcolm:] Yeah. Minus two is at minus one.
[speaker003:] Minus one. Mm.
[Malcolm:] Which is unfortunate the way we've got it numbered but that'll do. Minus Yes?
[speaker003:] Yeah....
[Malcolm:] M minus one is at minus one,... as well.
[speaker003:] Mm.
[Malcolm:] Nought is at one....
[speaker003:] Mm.
[Malcolm:] One is at five.... Two is at eleven,... which is about there isn't it?
[speaker003:] Mm.
[Malcolm:] Have I gone I've gone over slightly have I? [whispering] Never mind. []
[speaker003:] [LAUGHTER]
[Malcolm:] And three is at nineteen.... Now the thing about this is look y y This is alright, you can get this curve in here.... Dow down there, and turn around so you can get through and get the curve. Now when you get down to one you do not suddenly t put a straight line across there.
[speaker003:] Mm.
[Malcolm:] You carry the curve on round... come back like that.
[speaker003:] Mhm.
[Malcolm:] So in fact one of the things that you Cos usually you're given more points here so you can get a nice curve on both sides so you get a nice... Don't go from there to there straight across.
[speaker003:] Mhm....
[Malcolm:] Cos you'll lose marks for that. You want a nice smooth curve. Does it?
[speaker003:] Mhm....
[Malcolm:] Yes.
[speaker003:] Mm.
[Malcolm:] We'll do other things, but it's gotta be done with that... later. But what I'll do is I'll set you some work on stuff you need to revise and so forth. And we shall [tape change]