Okay, so we are progressing through the benandden paper. Let's Carry on. What at. So we've done this, we've done that. We've done that. We've done that, we've done that. But I think possibly you did a bit more work and. I think we did do that and we did that. So let's start with question 21 right at the end of the paper. And I think we're going to do some topic intensive stuff today after we finish. So pointing the area in the perimeter of the following shape, all of the angles are 90 degrees. This is a perimeter like all over Yeah exactly. Perimeter all around the outside of the shape. Yeah, very good. And the area the what's it inside the shape? The space inside the shape. And that's way we've done this. We realithat's 5 cm. Good. Good, really good. Well done. Absolutely right. A perimeter is 43. Yeah p should we just double check that? But something slightly different. So if we start from here and go round that way, we know what our starting point is. So it's six plus five. That's eleven plus three, 14 plus 721 plus 6:27 plus 5:32 plus four, 36 plus two, 38 plus one, 39 plus five. 4:44. There you go. But good youstill get Marks, Isabella. As long as you make sure to draw this, you're still going to get Marks. Well done. And what about the area of the shade? Keep these up because I think they're pretty pretty useful actually. It's just an easier question. Now, if this is 12 cm and this is 40 cm, the area of the shape which is inside the shape is very, very simply just 40 times twelve, which is. Four, 180, and importantly, centimeters squared. So you don't have to do anything too complicated here. You're just dividing it up into rectangles. There are different ways of doing this, but one quite easy way of doing it might be to divide it there. To divide it there. And can you see that you're working out the area of three different quadrilaterals and then you just add them all together? 64. Good. The only Yeah, no, no, no, that's all perfect. Very, very good, very nice. Fantastic 64. And what's your unit here is at 64 cm. So actually you would have gotten three Marks for that. So perimeter was 44 cm. And what would you put for the area? Really important, the unit here as well. Degrees? No. So degree is when you've got an angle. This is degrees. You know, it's like 62 degrees for something like this area. We did it just a second ago. Centimeters squared or meters squared or whatever you've decided to measure. The important thing is that it's squared. That makes sense. So 64 cm squared, if you're doing a volume question, you know, if we if we make this three dimensional, what would the unit then be if I. Draw some diagonal lines and I say, what's the volume of this shape? Good cm cubed, we say, or cm to the power of three. Yeah Yeah so it and it's because Yeah, it's different dimensions. So a straight line is centimeters. Something which is sort of two dimensional is centimeters squared and something which is three dimensional that has you know three different it's going in three directions. Well, I suppose I don't need to do it again. Like this is centimeters cubed. Good, let's keep going. So question 22, nice bit of algebra to solve. Now try not to panic here. You do know how to do this. It just might take a little bit of time. It's a three mark question. It's not meant to be easy. See if you can chip away at it and come to an answer. And I do it like this. I'm a little bit confused by that. I mean, look, with this, this is the same thing as what's three times eight. 24. And what would this be? Six -66Yeah if you can see that this has got nothing to do with this is not equal to eight minus two x over three. They're completely different things. So let's do a warm up question here. Let's do a much easier question. Forget about this for a second. Forget about this question here. If I say x plus four over seven is equal to 20. Remember how we solve this. So remember, whenever you see an equation, equation, remember, just means something which has an equals sign. So if you've got if they're saying what is x and there's an equals sign, that's an equation, that's all that means. So whenever you see an equation with fractions, remember. This is really important that you can multiply by the denominator. Now what does denominator mean? Can you remember Isabella? The number and down exactly to sort of get rid of meaning to eliminate to cancel the fraction. As long as you multiply the whole equation. By that number. Now again, we before we start, let's just again practice that idea. If I say what is x over three times x, sorry, times three, I meant what would that equal? X absolutely. What would two over seven y squared times y squared equal sorry times seven y squared? Good. So you get that idea. When you multiply by the bottom, you get the top. It works every time. You don't need to worry too much about exceptions to the rule. There are no exceptions to the rule. If you have a fraction and you multiply by the bottom, you get the top every single time. It's just the way it works. We can prove this by using numbers that we know, you know, two, one over two. A half times two is equal to one. And we know that's right, a half and a half is a whole is one. So we're going to use this idea to solve this equation now. So remember that if we times by the denominator, if we times the whole thing by seven, what do we end up with? Plus four equals. One. Hundred and 40 exactly 140 and what is x therefore going equal? 136. Absolutely good. So this isn't an easier question that we got there, but the principle is the same. It's just there's a little bit more work to do now. So we're gonna to start here and we're going to do exactly the same thing. We're going to choose one of the fractions to get rid of. It doesn't matter which. We're going to have to do both of them at some point. So decide, go, okay, which fraction am I going to get rid of here when we get rid of the first one, the second one? Whatever you want really doesn't matter. Okay. The second one. So what are we going to times the whole equation by. Good. And remember that in a question like this, sometimes it's quite helpful just to remember what to do, to think about what two is as a fraction. How would you express two as a fraction? Two is equal to what over what. Or over a two which is equal to two over 12 over one right so that this is it's kind of useful because what we'll see in a second so I'm going to write times two or if you like two over one they're the same thing they're just different different ways of saying the same thing but when we times by the fraction that it's really helpful because we can see okay so actually I'm going to times the top by two but not the bottom because it's two over one. So let's do this now. So starting off here. This is what we're first doing and this is when it's really useful to think of it as two over one if you know what I mean because now we know what we're doing. We're doing eight minus two x over three times two like that is going to give us what. We earn 40 minus four x over three. Say the first part again the top. 14 minus four x where did 14 come from? Two times eight. Is two times 8:14. 1616. Thank you. 16 minus four x over three is perfect. Well done, Isabella. So we can write that for the, for the first bit. Fantastic. What about this second bit? Now? What's two x minus three over two times? Two. X minus three. Absolutely. Very, very good. Now just notice this is when life gets a little bit tricky. But can you see that we've got a minus here? It says minus. So what we have to do is we have to imagine actually what we're really doing is we're mining all of this, the whole thing. We're minsing everything. So when we times it by two, we get rid of the fraction. We got to remember that we are still minsing everything, so I'm gonna to put it in brackets like this, two x minus three. And finally, what's four times two? Eight. Good. Now before we continue, let's just tidy this up because it's a little bit messy at the moment. If I get rid of the brackets, what would this go to? What would what would if you ever have a minus outside of brackets? If I have something like minus x, minus y, do you know what that would equal without the brackets? Minus two x plus three. There was a little bit of a guess that felt like remember that whenever you see a minus, it's a minus one. We don't write the one, but there's a minus one there. So to expand out the brackets, you're doing minus one times x, which is. Minus two x. What's one times x or X X? So what's minus one times x, minus x good, minus x and what's minus one times minus y. Why? Yeah, absolutely positive. Y, good. So I think you were actually doing the question on the right. I realized, so what's this going to multiply out to again, minus one times two x, minus one times minus three? X plus. Yep. Can you just say that again? Pay real attention to the positives and negatives. Minus two x plus. Well done. Plus three exactly is equal to eight. Good. So now we're just doing exactly the same thing. What are we going to times by now? Three good going at times everything by three. What does this give us? 14 riders for 1616 minus four x. Okay, just just be careful there. So if you want to if you want to do it because we're times ing by positive three, you could say plus three times minus two x because this is not negative. Can you see that's positive there? You're not times in by minus three, you're times in by three, which is the same thing as positive three. Remember, plus positive minus negative, they're the same thing. We learn them as different when we're Young, but they are the same thing. So. But but it's easier just to do it one by one. Isabella, I would I would suggest just think what's three times minus two. Minus six x good. What's three times three? Nine good excellent and what's eight times three? Four. Good. Let's see if you can take it from here and solve for x. Just be careful if that minus one is correct, just double check minus four x, minus six x. Think about it. Think about what minus four minus six is. You're starting at minus four, you're going to minus six. Yeah. So imagine zero is here. We've got minus four here, and we're going minus six in this direction. What's here? Good. Minus ten x. Thank you. Good. Really important. The positives and negatives. Got to sppay special attention to those. Good whono, so remember, don't do it all at once, Isabella. You had it and then you lost it. So if minus ten x is equal to minus one, you can say ten x is equal to one, right? If minus x is equal to minus three, then x is equal to three. So do that first and then you're not going to go wrong. Then you can say x is equal to positive one over ten. If you miss out steps, you're more likely to go well, not bad though. I think possibly we could do a little bit more work on that. So they've got one last question, but to be honest, let's not bother about this. I'd rather do some some more kind of topic intensive stuff. So we've got a list of things that I think would definitely benefit from from a little more practice. So let's start let's start off an nth term because I think this is one that you can get really quickly. So let's imagine that we have a sequence eleven, 18. 25, 32, 39 etc.. So this is a sequence. Sequence just meaning a group of numbers that are following a kind of logical pattern. So you look at that and it's a always a wise idea, even before you start a question to sort of think, what you know, what is this? What kind of sequence do I have here? This one is a relatively simple one, I would suggest. What would the next number here be? 46. Good. 46, absolutely. And in doing that, you identify fies that we're going up in regular intervals of seven. So we're adding seven every time. Now, what they asked in that last paper that we did, and what's a really common question, is they can ask you, what's the end term of this sequence? Now nth turn. This is again, something to revise and learn. So th turn. Is like the code for the sequence. Do you know what I mean by a code? Code and Chinese. Like this word. Yes, makes sense. So it and so and after. You have worked out the nth term. You can work out any parts of the sequence really quickly. So here's how you do it. So the nth term, what we're going to use, first of all, is this number here. So you can see that it's going up in sevens. So the first thing that we write is positive seven, and I'm going to write n. So just whatever the difference is, if it's positive, write positive seven. If a sequence is going down, write minus three or minus eleven or whatever. That's the first part. It's really, really straightforward. Second part, you work out what this number here is. What would this number here be? Good. So seven n plus four is our answer. That's it. Easy as that. So just the difference. And then you work out one to the left of the start of the sequence. You're done easy. Now what this allows us to do is I can you can get a question, something like what is the hundredth term of the sequence, right? So what this means is at the moment, we've been given the first term, we've been given the second term, we've been given the third term, the fourth term, the fifth term, and you actually already worked out the sixth term. And that was quite easy. But if we're going to work out the hundredth term. You know, it's going to take us a long time. We might come up with a clever way of trying to do it without writing 100 numbers, but it's risky. The best thing to do is to work out this, the nth term, and then you can work out anything. So to work out the hundredth term of the sequence, all we do is we use the nth term where n remember, n is those these little numbers at the top. So when at the hundredth term n is 100, so all I do is I go, okay, so seven n plus four where n is 100. So that's going to be seven times 100 plus four, 704 like that. And we know now that the hundredth term of the sequence. 704. Why is plus who I forgot? It's this number here whatever that number it there is Yeah so I think rather than you know keep continuing to explain it it's it's best if we just do lots of examples and hopefully the the method is something that you'll you find easy to remember. So let's just do another one straight away. Let's do 15, 18, 21, 24, 27, blah, blah, blah, blah, blah. And again, I want you to work out the nth term of the sequence, and then I want you to work out the thousandth term of the sequence. You only need to do one here. Don't don't worry about all of this. So this this here. No other way around. So you're working out the difference. What's the difference? Yeah, good. There you go. Three n plus twelve. Perfect. And what would the thousandth term of the sequence therefore be? 3000 inch per absolutely right. Let's do another one. 42 4051. 6069. 78 again tenth term, two hundredth term. You're doing it the wrong way around. It's nine n plus 33. So try to change that. You've done that twice now, Isabella. Try to change your understanding of it. Nine n plus 33. But that's correct. Well done. Nine n plus 33. You've got the numbers right. Wrong order. Good. So the two hundredth term would be. Yeah, absolutely. 1833. Good. Another one again, just so that this really becomes second nature. 14. Let's do one that goes down this time eleven. I can't count. Eight, five, blah, blah, blah, blah, blah. Same thing here and th term. Hundredth time. Same rule, nothing has changed. Good. But is it three? Is it three? Isabella being really specific? This. Thank you exactly. Very good. Yep. And just again, be careful that you're slightly rushing your work. So it's minus three times 100 plus 17. So that's -300 plus 17. Think about what that is. You're often getting confused with negative stuff. Good. Yeah. Yep, good -283 perfect. Very very good. Can let's do another let's do three eleven. 19, 27, 35 et cec etc.. Tenth term. Three hundredtime. Good. Good. Well done, Isabella. Absolutely agree. Have a think about this. Nearly good. It's really, really nice work. Okay, so that's making sense. You've got that. Just try to remember that now. So the the difference is the number that you put in front of the end, the and then you can just use this to work out anything. They can also as well as asking you sort of like what the three hundredth term is, they can also say what term of the sequence is, for example, four, 23, or am I sorry, do I mean that I mean 4203? Now just notice that when they ask you this, what they're really getting at is what you've worked out as the nth term. If we imagine n is up here, so this is one, this is two, this is three, this is four, this is five. And you actually just worked out incredibly what 300 is. You worked that out a second ago. It was 2395. So now what they're saying is, what term of the sequence is 4000, 203? In other words, what does n equal up here? Hundred? Yeah, so it was 300 for 2395 and they're saying, what is it here now? Any idea how you would do this? Multiply. I don't know. Well, again, you have the code. You have the code. The code, remember, is this? So that's going to get you anywhere. So all we're doing this time is we're trying to work out what n is rather than working out what the sequence is. And this one here, they basically say, you know, n is equal to 300. So we do eight times, 300, 2400 minus five, 20 395. But this one here, they're saying, what is n or what does n equal when A N minus five is equal to 4203? So you just write that out you go, okay, my end term is 8:10 minus five, and I've been told that it equals 4203. So I'm going to work out now what n is. So I add five to both sides and I would get eight, ten is equal to what? 400. 4000I mean, it's going to be 4203 plus five, right? More thousand. 208 good and what is n therefore equal to? Just be careful with that. What does 42 divided by eight equal? Oh, no. Oh, sorry. You're absolutely right. I got the question wrong. You're you're absolutely right. Forget what I'm saying. Good, good, good, good. You're doing it perfectly. 526. Nice. Very, very good. Absolutely right. 526. Excellent. So you've just worked out this here is five to six. So similar similar to the last question, but rather than working out rather than being given n, you're trying to work out what is n when you're given the number. So again, let's just practice this and we're going to do another. Let's do 20, 27, 34. 51, blah, blah, blah, blah, blah, and same thing. Let's have nth turn. Let's have the two hundredth term, and I'm going to say what term of the sequence is, what should we do? Maybe. 500 and. 47. Yes. Good. Careplus three. Good. And this is a bad question by me. You're doing it perfectly. Good and this is gonna to go on and on and on and on and on and on and on so we're never gonna to stop there. So actually that that was a mistake by me, but it's it's a it's actually a really, really useful opportunity to say 574 is not in the sequence. This is actually what we just worked out. If you if you're getting a decimal answer, what that means is the numbers not in the sequence. So I'm going to change the question now to something that does work, because I think I had a bit of a ditsy moment there. Let's just erase this and I'll come up with a question that does work and let's change it to. 573. Good. Yeah, absolutely. Eightieth time and now we can say yes, 573 is in the sequence. It is the eightieth term. So this is nth term. Hopefully that's making sense. And I think it would be a good idea today, maybe after the class just to just to make a quick note of how that works because you've got it now we're gonna we're gonna to return to this at the beginning of our next class. And I don't think we need to spend too much time on it. I think we can move on quite shortly after, but you know you can you can end up with six Marks or something for for a question testing this topic. It's pretty it's pretty simple, isn't it? It didn't take you long to to learn it, so make a note of how to do that. Maybe I'll send a question or something if you want to practice it just to jog your memory. And Yeah, I think we'll park that for now and we'll move well, we'll return to it at the beginning of next class and then we'll do something else after. All right. Well done, Isabella. Take care of yourself and I'll see you next time. Bye.