GCSE MATHS: Iteration Exam Practice LIVE LESSON 14 EDITED

hey guys just before we get into the lesson I just want to let you know that you can catch these Live on YouTube uh every Tuesday from 6 p.m until 7 P.M in the UK time and that means you can be more interactive you can ask your questions and they'll be answered okay so without further Ado let's get into the lesson hey guys and welcome back to another one of the GCSE tuition lessons so today what we're going to look at is uh so we're going to start off just finishing up the content that we've uh started so we're just going to finish off with iteration and then I'm going to talk to you a bit about exam technique so I know you guys are going to have some mocks coming up and you know maybe you have exams coming when you watch this whatever I'm gonna go through some exam techniques and some tips in terms of that so let's just get started with the iterations so with iteration uh normally questions are about three or four marks and they're really really not too bad I'm going to show you a trick on how to do them in about 10 seconds they'd be able to uh manage to get all the marks quite easily hey Evie how are you so yeah with iteration not too tricky essentially all iteration means is you do something over and over and over again okay so what we're going to do is start with this question so it does the number of rabbits in a field two days from now is PT where P0 so that's the original uh not too bad uh unfortunately I came down with another fever over the last few days um because one of my roommates Works in a primary school and children are disgusting creatures and yeah so she put it home like a rabid dog I'm not too bad now though to be honest uh the fever's gone I have a bit of a cough that's about it um I'm actually feeling pretty decent though not too bad not too bad but yeah that's also why I'm in a t-shirt because I didn't go to work today and uh I just didn't put on a dimple on shirt and tie thank you very much I really appreciate that thank you so uh what's the end of today just in case you didn't hear is we just do the iteration because that's like the last major topic we haven't done and then we're going to some exam uh strategy stuff so exam tips and everything because I know that you wanted me to go through how to think logically and stuff like that so I have a few tips that might help you for your exams so essentially with iteration is they always give you an initial amount here that's what the zero stands for right T days from now so zero days from now means today and then to work out the next day is you take the current day and sub it into that formula right so what they probably teach you to do is that P1 so one day afterwards is you do 1.15 times 220 minus 20 right and you put that into a calculator work it out right so 1.15 bracket 220 minus 20. 230 and then you'd you'd keep going right so this is one way to do it and I'm doing it the long way just this once so I can show you a shortcut which again I know some of you have already expressed that you know what to do and then all we do is we change that three and then we're going to keep going 241.5 and then lastly P3 also now that a few more people are in I do have to apologize to you guys because the lesson ended quite abruptly last week so I didn't end it but one of my roommates wanted to watch the uh World Cup match I can't remember if it was England or not and accidentally turned off the internet instead of turning on the TV hence uh the inside cut off and because he didn't realize in time by the time he had turned it back on and the stream started up again everyone else was gone because uh YouTube automatically closed the stream so I do apologize for that so this is like the legit way to do it I guess you would call it however and again this is three marks right and it didn't take me three minutes to do so it is definitely doable this way and again your answer would probably be 255 you just round up right because they're rabbits you can't have 0.725 of a rabbit however there is a quicker way to do it and with these three markers the reason why I love iteration so much is because you have three minutes to do a three marker if you can do this in about 10 seconds it means you have three extra minutes to spend on other questions on the questions that are a bit harder right so here's how you can do it quite easily start with the number that they give you so 220 and just hit the equals button okay now this ANS button here saves your last answer that's what a s stands for it stands for answer so what you can do is now Sub in instead of p t we're just going to put the answer button minus 20. now that answer is currently 220. now if I press equals it gives me 230. press equals again it'll give me my next answer 241.5 press equals again it gives me 254.725 and again as long as you write down all of those numbers you get the full three marks and I did it in a few seconds and again I know some of you have learned this so we're going to go through some more of the questions see if we can get it a bit uh more tricky ones and then we're going to go through the exam technique and obviously you can also ask me any questions that you may have so let's get started so again number people living in the town is this workout number of people three years from now so again P1 equals P2 MP3 so the method box just you get given for writing P1 P2 and P3 and what they are so it's quickly work it out so we have the initial thing is 55 000. right so that's now my answer then I just typed in the equation as I see it except instead of the variable I'm going to put the ANS button minus 800 and now if I just press equals I get that and then all I need to do is write them down and those are my method marks just in case something does go wrong which it shouldn't but you know better be safe than sorry and you don't need to round up until you get to your actual final answer okay so even though it doesn't make sense to have 0.78 of a person we're going to round up just right at the end and then 57553 blah blah blah so three years from now you're going to have fifty seven thousand 553 people fairly easy so far I think now there are slightly trickier versions uh this isn't actually one of them this is actually okay find these three values again just because they've changed the letter and just because it looks a bit more complicated doesn't actually make it any harder x0 is three says press three and then equals right now I rewrite this formula but instead of x n I'm writing A and S so three Plus fraction 9 over A and S squared and now let's press equals three times and that's it okay so you can save so much time doing this which you can then use on the hard questions you know some of the uh end questions that are quite uh sorry I'm a bit foggy at the moment on a lot of medication some of the questions at the end are five six markers that require a lot of thinking normally it takes you longer than five or six minutes so these questions are great for kind of saving that time and bringing it back so instead of having to do a mark a minute you can save some more time hey Lila I always find the end easier than the middle uh yeah sometimes they can do that so actually I was looking for a few extra questions that I can use as an example for my exam tips I'm going to do a bit later on and I did notice that the last questions weren't always the worst but a few questions before the end were so yeah it's good to say to actually save as much time as you can whenever you can right and again we have these so I'm assuming iteration is fairly easy at the moment but there are some difficult bits I promise where they also ask you to rearrange these formulas and do all kinds of uh funky stuff like that so we will get onto those in about two seconds and again if you have any questions do let me know uh also just in the back of your mind have a think about what you want me to do next I was thinking from next week onwards to start doing exam papers but if you have other topics you want me to focus on then please let me know okay but uh yep so five over four we should probably write it as right and oh god let's write it as a fraction actually just to make my life a bit easier so I think these types of questions are okay for now right so why don't we go on to something a bit more interesting but in terms of iteration that is exactly what I would do right just use the ANS button and I did these in seconds right it's not too bad however you can get to different slightly different uh Parts but what you should know and again if you're doing a level maths this is going to be quite useful as well um but just for the exam is that the whole point of iteration is to find solutions to equations okay so if they ever ask you and there's a question coming up on this in a second like what are X1 X2 and X3 they are approximate solutions to the equation that you rearranged so this one in this case okay so it is a very useful thing to do and it happens a lot in kind of maths and science so just so you guys are aware there are certain mathematical equations that we can't solve exactly so instead we use iteration to find really good approximate Solutions so for example wanted one equals two right I know high level but let's say if I didn't know one and one was two but I could find an approximate solution that said it's [Music] 1.999999987 right that is a pretty reasonable approximation right that's the kind of thing that we actually do in real life so again personally I did a degree in physics some of the problems we have we can't solve them completely right and computers aren't powerful enough to get an exact answer so what we'll do is we'll get something that's corrected about five decimal places because that's correct to a 100 000th of a degree which is really accurate and we don't need to be more accurate than that so iteration is really useful in real life it's used all the time because to be honest some math equations are too hard for us to solve and they're too difficult for computers to solve as well so I think we'll leave this question because I think again it's just the exact same thing you use it three times and your final estimate is going to be an estimate to this equation okay same thing now this is the question I was coming across here so we'll do this question here said again start with 2.5 press equals that now makes that my answer button now rewrite this formula but instead of writing the X N bit I'm just going to write answer and again it doesn't matter what letter it is just whatever the variable is replace with the A and S button and I just keep pressing equals nice and easy and I'm going to leave this in terms of a fraction because I think it's going to get quite messy otherwise yep so again just have to think about what kind of questions you want to ask me at the end of this but I do have some exam tips I want to share with you so this is the part part B that they can sometimes ask you right explain the relationships between those three values and this equation here ah so they've actually made this a bit more tricky sometimes they give you this equation sometimes they don't but in this case we have to show that this equation has something to do with this equation okay so this is what I mean by sometimes they need to rearrange so this really tests your algebra skills so what I need to do is show that this equation can be written like this okay so if I I'm going to write out both equations just just so I can do it X cubed minus 2x squared minus 5. 0 and the equation I want to show is that one okay 5 over x squared called two okay and again this can be quite tricky because it's not rearranged in the way that you might expect all the time so if we have a look the five and the two are positive which means these two things must have been moved to the other side right so start with that so again this might help with the whole thinking logically thing that you mentioned before right so I'm thinking logically by looking at the answer look at the signs so that tells me what the sign should be in my answer as well right now if I look on the left hand side there's just X whereas on the right hand side there's an x squared so I'm going to divide both sides by x squared in order to make this x equals instead so divide 3 by x squared and I get x equals two plus five over x squared so now if I just rearrange this slightly that looks exactly like this does right so now that I've shown that those two equations are the same I can now say that x 1 2 and 3 so X1 X2 and X3 our approximate Solutions to x cubed minus 2x squared minus 5. equals zero and that's it okay by rearranging these formulas can get a bit tricky in terms of the algebra because you need to know what they're thinking about and again it does come back in a level as well but so far we're okay with that I'm assuming we're finding it a bit easy would that be fair to say good yeah is this worthwhile like is this is a good thing to go through because again if you're sitting there bored not really caring because you know all of them that's not good but is this okay because again this is like the last major topic we need to go through really oh perfect approximately it means close to yeah that's exactly what it means yeah it doesn't mean exact so approximate means it's roughly correct okay so for example if I said uh PI right it's infinitely long but I could say it's approximately 3.14 right because it's close to 3.14 but it's not exactly 3.14 or if I were guessing your height I might say you're approximately five foot six or something it's not exact but it's an approximate so another kind of question they can give you so show them the equation this has a solution between X is 1 and X is two so again this is a common thing that they ask you again comes up again in a level etc etc so what you want to do here is you sub in the values for X so when x equals one what does the equation give you okay and well I don't need to calculate for that 2 times 1 is 2 minus one is one minus three is minus two and X is two okay so 2 cubed is eight times two is sixteen minus 4 is 12 minus three is nine cool so here's where things get a bit more interesting okay when you sub in those two values if there is a change in sign then there is a solution between those two values okay so if this first value is negative and the second one's positive or the first one's positive and the next one's negative there exists a solution okay so what you want to write exactly is change in sign and I'll go through why in a second of okay so are we okay with why that's the case or would you like me to explain this have a think for a second and then let me know if you would like me to go through it as well in fact I'll probably go through it anyway nope okay so we want to find a solution for when this equation equals zero right equals zero now if I were to draw this little curve out I actually don't know what it looks like not completely but since it's a quadratic and it's positive it's going to probably look like a u like this yeah I don't know what the root side doesn't matter I don't care when it equals zero I'm basically looking for the x-intercepts yep so the word roots is the same as Solutions okay so Roots solutions they all mean the same thing oh thank you Evie I made a mistake I told you I'm a bit dazed at the moment yeah it's a cubic graph thank you well spotted uh so it's going to look let's say something like this right I don't know exactly what it looks like but it doesn't matter okay and I have these points here thank you for pointing that out yeah I do feel a bit dazed at the moment my apologies okay so we want to find where equals zero which is where it crosses the x-axis right which yes are the roots or Solutions now I've subbed in x equals one and x equals two now when I subbed in Norway's hydrocarbon welcome now when I subbed in X is one I got a negative value so I got a value that's somewhere down here right and when I subbed in x equals 2 I've got a positive value now because my graph is a continuous line if it goes from negative to positive or positive to negative it has to go through zero okay every function you deal with except reciprocal functions always our smooth curve they're called continuous curves okay which means if there's a change in sign so if it goes from negative to positive or positive to negative it has to go through zero so look over here if I subbed in x equals whatever this is I'll get a positive value then I'd get a negative value and it does go through zero between those two values okay does that make a bit more sense so hydrocarbon I don't know if you can pick up what's happening because you've joined slightly late but essentially one of the iteration questions they can give you oh that will make sense okay uh EV is that okay so yeah perfect okay I'm glad that makes sense now and again thank you for spotting I drew the wrong graph and lastly so again these are the hard questions in my opinion normal iteration I don't think is that difficult I think you guys are completely fine with it but when it comes to rearranging and then some of the slightly diff not difficult different things I can ask you it can get a bit tricky so so the equation 2x cubed minus x squared minus 3 can be rearranged to give this so again we need to use our logic now I said before that we can look at the signs right so if I look at the signs the 2 is positive so it needs to stay positive the three is positive this one's good uh what do you mean by that uh anyways uh the three is positive so I need to make that positive somehow and there's a square root okay easy to get yeah so since there's a square root this thing on this side is going to be the uh the x squared here so I'm going to move that x squared to the other side so now I have x squared equals 2x cubed minus three okay so what I'd really want to do next is so again think continuously think about what you need to do so actually what might have been better for me to have done is to because the 3 is positive move the 3 over here and the x squared over there and then not divide by X cubed not divide by X cubed because again we need to square root it at some point right so what we're actually going to do is factorize out an x squared and if you have a look I now have the denominator of that fraction so I'm in the money okay so even though I made a slight mistake in the beginning I constantly think okay look at the answer does it make sense does it make sense does it make sense okay so now if I divide both sides by 2x minus 1 I'm going to have x squared equals 3 over 2x minus 1 and then all I need to do is square root both sides and I get three over two x minus 1. yeah you can if you want to you can sorry you can constantly just kind of rearrange stuff whenever you want um obviously so with your teachers I imagine that they tell you that there's one specific correct way of doing things but they just want you to be really efficient right which isn't the worst thing to be but you can do whatever you want so long as you follow the rules of bats which is a good thing right so uh we'll do the iteration because a few more people have joined so I'm going to show you the little trick that I have so again these are normally three markers and you can do that in about 10 seconds so what I do and I imagine some other people do as well is I take x0 which is the number we start with right so one and I just press equals so now the ANS button just gives me that one because the anus button saves your previous answer okay and then all I do is I rewrite this exact same formula but instead of the variable I'm going to put the ANS button so square root 3 over 2 A and S minus one now every time I press equals it's going to give me the next iteration right and it says use it twice so if I just say X1 equals root 3 . then if I press equals again I get what the next iteration would be so in this case I use it it just says use it twice to find an estimate for the solution of so your estimate is going to just be 1.103 and that's it nice and quick we probably don't need to do another one of these so let's do another ways okay so this is something that some people get confused on okay so you know how before I said you sub in the value and you should get a difference in sign it only works if it equals zero it could be rounded to the newest whole number yeah it could be so uh nearest whole number or like one decimal place absolutely if you want to show this equation has a solution between 0 and 1 what we have to do is make it equal to zero first okay so you need to do just move it over to the other side now again this obviously seems like an easy step now but you'd be surprised at how many students forget that in the exam because again we're looking for a change in sign I won't get a change in sine if it equals anything other than zero okay just be really careful with that then again we're just going to sub in x equals zero that whole thing is going to be zero plus zero minus one is going to equal no Graphics tablet it probably looks like I write with the mouse because my handwriting isn't the best but one plus four is five minus one is four yeah I do I write with the graphics tablet so my handwriting isn't as bad as it could be and yeah so again we have a change in sign and again you'd write implies thank you today it's not as good as it could be because I'm feeling a bit uh dizzy so I'm slightly ill no honestly writing I know some people that can write with the mouse pretty pretty well but they're very slow and I don't want to be slow and my handwriting isn't the best but so I wanted to make sure it's still legible for you guys but yeah I use a graphics tablet and show that you can be rearranged okay this this shouldn't be too bad uh one thing to note though is you're not rearranging for X cubed right because the X cubed's actually over here it's actually just rearranging for X so again look at the signs there's a minus X cubed everything's divided by four so using those things together you can probably see we want to get 4X by itself foreign by four but you do need to show the steps okay so do it in a in a couple steps just to make sure you get the uh marks that's it um also I don't know if you guys saw I did upload that video about the uh breakdown of exactly how uh the examiners kind of Mark your work and the criteria that they have and it is the same for all exam boards and all levels if you haven't I can go through that again at the end of the lesson as well that's completely fine because we're actually almost in fact this is the last question so yeah we can definitely do that um so yeah that's it with iteration so I want to go through some exam technique stuff and obviously if you guys have any questions you're more than welcome to also if you have any suggestions for topics so now that I've got a few of you here um we've basically covered all of the major topics right so we've done 40 this is the 14th lesson we've done pretty much everything I can start just going through exam papers from next week that's completely fine or if you want me to go through something else I can also do that so what I could do is like get the last few questions or every single paper and just do those or whatever so if you guys want to have a think just message as soon as you know what you kind of want and then we can go through that but in terms of the kind of assessment objective stuff did you guys see that video I made or do you want me to go through that now Mega heart exam questions okay well I might as well write that down you saw okay okay good I hope that makes sense um by the way that changes for a level so GCSE is everything standardized so every exam board yeah we can do that that's completely fine um what I also want you guys to know is that because you guys are probably doing different exam boards if I did like an ed Excel paper it is going to be useful to you right if I did an AQA paper but you're doing something else it's still useful to you because every single exam board is judged on the same things right in every exam paper you do the same content okay so OCR cie all of that stuff is it's all completely the same so it's different for a levels but for GCSE all of the content is the same across all exam boards and there's no difference to the breakdown or anything so realistically speaking if I gave you like some random exam paper it's all going to be useful but we can go straight to exam papers Mega heart exam questions good and did you guys have any questions about that video uh in terms of that foreign exam questions I like that um perfect okay so add a request to kind of go through how to think more logically and also like how to like approach questions and things like that so one thing I will like to say is more like it please okay yeah definitely I can do that um with questions and I know this is probably advice you've heard before but you don't 100 believe it yet is can we go through 10 Okay so turning points and roots of quadratics with okay sure yeah we can do that so when you go into an exam question right every piece of information they give you is useful right they don't give you stuff that's not used in the question so one thing to be really clear on is to yes read the question but think about why they've given you the information they've given you right because it's almost always in fact it is always going to be useful for the question so what I have is I have some collection of exam questions open and these are a bit tricky so yeah let's do it let's do it uh should I print it over let's just do it okay so ignore the Thousand tabs I have open I'm more organized than this I promise no I'm joking it's because I bought a lot of like work to catch up on because I was ill for the last few days so if I make this slightly bigger so yes this is an AQA question right and this is probably one that that would catch quite a few people off I know I like functions too so they give you a graph x cubed plus six is translated four units to the right it's the translated graph has the equation y equals f of x work out f of x give your answer in the form right so if you have a look at this oh why can't I draw on it I'm assuming if I saved it I could nothing one second so if I just save this on my desktop for now can I draw it of course okay it's fine I'll print it over so we can actually write on it but it tells me it's been translated four units to the right and it needs to give this answer in the form so a lot of students will approach this question and a there'd be kind of apprehensive about it they won't know how to answer it but if we take it step by step every piece of information is useful okay they've told us the original equation I mean that's pretty obvious that's going to be useful and it's translated forward to the right so what I would do is think about okay what does it mean to be translated four to the right well okay you know what let's I'm not going to print it so I'll just uh screenshot it whoops yeah so what does it mean by translated four to the right well what they're trying to test here is if you remember that for it to be four to the right it needs to be inside the bracket because it's affecting the x-axis and that means you need to put the opposite number in right because when it's inside the bracket you put in the opposite value right so you flip the sign on it does that make sense so far so now what I need to do is I need to look back at the original and think how can I make this happen well all you need to do is replace x with x minus 4. so then I have x minus 4 cubed plus 6. now that would actually be one answer but they want you to give it into this form here now that first step is where most students messed up on on the question okay because they they didn't make the link right some of them did make the link but instead they did this or they did this because they've forgot about it or when they subbed it in they did something like this they did X cubed plus six minus four so again every piece of information they give you is 100 useful so what you need to do is take stock of what they've given you right this is useful this is useful so think about but this how can I use it that's what you should be thinking about in an exam now one other piece of advice I'd give you is if you can't think of an answer in 10 seconds skip the question and go on so in every exam I've done I will read the question by the time I've read the question I give myself a few seconds if I can't think of what to do I go straight to the next question and then when I get to the end of the paper I go back to the beginning work my way through again okay because you can't waste time in exams they really don't give you enough time right personally I think they should give you an unlimited amount of time because what's the point in rushing you right but you don't have much time so just keep rolling through it okay I've been through papers three times and on the third time I'll do a question right because again I'll look through the paper I'll get to the same question let's say question two I had to skip the first time I come back around I look at question two ten seconds again nope can't think of an answer go to the next question and on the third attempt then I'll spend five minutes on it because I know that if I waste time on that question I'm going to lose more marks than it's worth the other thing to realize is that your brain has two halves right you have a conscious brain and an unconscious brain if you read a question right and then go on to another question your unconscious brain will start working on a solution and I've had this happen to me many many times how many times has it been where you've like like you know when you're in a shower right and then you think back to an argument you've had with someone and then you think about all of like you realize all these good things you could have said it's because your unconscious brain is working on that problem right and it's only when you distract yourself that you can think about it so it's a similar thing to that yeah it's a similar thing right I've had times where like in my degree where I'll sit looking at a problem for 10-15 minutes I'll walk downstairs and while I'm like getting a glass of water I'll go oh wait that's it right it's good to kind of let your unconscious brain have a garden for a bit so again 10 seconds that's a long time to be fair right to just be staring at a piece of paper so really really try and remember that um would everyone be okay with this so afterwards by the way this answer you would just expand the brackets and you're done and that's four marks but again students got a bit panicked about it the next yeah so the next one is a five marker again it's just a solving algebra question but a lot of students found this one tricky as well and it's probably because X turns up three times in the expression I know yeah so these are very common but again a lot of people find these tricks also what I want to point out right is it's good that you're saying you've done a lot of these because maths is just practice especially GCSE maths so it's good that you're asking me about logic and things like that because they are useful but truth be told a lot of uh GCSE maths questions just get repeated every year so you don't really need to worry about it that much so with this what would I do well I hate fractions so my first step in any solving question if there's a fraction is to get rid of the fraction right so if I think about it and how do I get rid of fractions well the easiest way to get rid of fractions is to times them by their denominator right so our Times by 4X plus 1 and I'm timesing the whole equation by 4X Plus 1. right that's one and then again I still have a fraction so I times that by its denominator so then I have five x squared plus three equals 2x or X plus one and now that's just a quadratic equation which you guys have solved a million times