Transcript for:
Essential Concepts in Math: Stats, Geometry, Trig

Third one. So I tell you all second first time. Those of you who are veterans tell us tell the first time people how is the Hello YouTube folks. YouTube folks should be seeing me all now. Um, you know, things will be in YouTube land. Very quiet, very deer cutesy. All right. [Music] right people being 3 minutes ago I'm trying to join the group just give me like a little 10 minutes straight cooking right through normal. It was normal. Oh, somebody now reach back home and send the receipt. I just trying to add add up a few people before I take off. Miss Nolin, why is he scene? near there and you're able to go um um live right now and just talk to the class about how we going to structure things today. Why is I just add up a few people right now my WhatsApp cooking I starting the recording. starting. So if you get it if you get a chance [Music] All right, thanks. I'm still sending a couple messages here for people who um did their last minute payments. We starting shortly. Um, somebody get invite but forget to join. [Music] [Music] All right. So, I was off for a bit. in Mac. Let's refresh. Uhhuh. I know what's going on. And I know what's up. That was my All right. So, and that's as much as I could help the last minute peers. Right now, the admins are on. So, if you have a friend who paid last minute and they're trying to jump, let them message the admin and we'll deal with that, right? Um, let me just miss like the first few minutes. So, it is time to begin. Let me grab my notes out here. This is going to be very interesting. Can you all let me know how the maths revision is going before I start? So a couple persons was messaging me like sir why don't come on live the guys the stuff that I did from last year and the year before still applicable all those um playlist that I have is the same mats I have so much stuff on YouTube already and it's really nicely done so what we also did Melini pointed out um just now just a second is that there's a document that you can find on the drive that basically indexes organizes my YouTube channel. So you can use that to find topics if you're having problem to search up and get the individual topics. Those of youall who did the paper tree, how was that? I haven't yet seen a paper tree. So, if you have a paper tree, you can feel free to send it to me. There we go. Wasn't too bad. The Lord. Praise the Lord. Starts coming. Wow. That's good. That's good. That is so good. You know that people can be a nightmare sometimes. All right. So I think it's time to begin. You'll see people filing in here all kind of time because some people have different engagements and it'll probably midway. Um so where am I supposed to do up to people? What is what Miss Nolini say I had to do again? What miss? Oh yes. Section one to three and then I have line and touch down and and the rest section one to three. I want to teach everything in l. So one, two and three. Uh all right. So let's begin. So actually I'll teach section one, two and three and then I'll teach section nine as well and that will take me three hour four hours about three or four hours and then we'll have a break. We'll have a break for like half um quarter of an hour 15 minutes. We don't have two long breaks cuz I'll meet up all our time and then Miss Nolini will do section six and section seven. I can't believe I give Miss Nolini section six and section seven. The two sweetest sections Mr. Nolini going to teach algebra and relations functions and graphs and measurement. Molini gain all the fun. I had to teach you guys the topics. All right. And then I'm coming back in in statistics, geometry, and then we going into overtime. So anybody who have any anybody who have energy after 900 p.m., you know, hit them a little extra like that be that be. So without any further ado, let's just jump into it. I think I'll jump into it. Um I hope Shora is on my WhatsApp. So I think she's on There's a couple people who messaging me frantically. They just be able to jump in. I still want to try and help them out. All right. So, let's go. Um, there's little there's little clickable stuff here. So, you're supposed to be able to click that and jump to it. Um this is a topical crash course. Let me just make sure YouTube people you all hearing and seeing of you all here on YouTube topical crash course means that I'm going through topic by topic. has its use. Um those of you who in the April crash course know that was a kind of pass crash course. So here getting to cover the topics both those of yall who managed to do both crash courses. Well done. So firstly you should know that this means the set of so the set of happy exam students. You know what I mean? It literally means the setup when you see that um this means and so on. So like if I do one two three or maybe two four six in maths this tree that's it means and so on. This means is a member of as opposed to this which means is a subset of uh we talked a little bit about that recently. There's a difference right? So um I could say Liam is Liam is a member of the set of students in class and if I have two subsets I have the girls and I have the boys um I can say boys is a subset of the students in class. You all understand the difference? So a subset is like another set that belongs to the whole set. A member is the actual element itself. So natural numbers 1 2 3 4 and so on. Whole numbers start from zero. You have to notice that this comes in the paper one. This comes in the paper one. Um and then integers. Integers include the whole number. So from zero and so on, but you also include the negative version to the natural numbers as well. Rational numbers are any number that can be expressed as a fraction. That's what a P over Q means. It looks a little strange. You're locked in, but that P over Q just means you can express it as a fraction. Any number that you can express as a fraction. So like 0.125 is actually 1 over 8. So most numbers in the world can be expressed as a fraction. Even if you're writing the most random number um any number that you can think of can be expressed as a fraction. If you write let's say 0.0014678. Well that's just 14678 over the relevant um degree like maybe one zero 0. So I could always express it as a fraction. The only numbers that we can express as a fraction are irrational numbers. And these are very specific numbers. Can you give me an example of an irrational number? Can you give me an example of an irrational number? Yes. Pi is the most famous irrational number. Pi is an irrational number. Pi is not 3.14. It's approximately 3.14. It's not 22 over 7. It's approximately that. No one knows the exact. So the decimal for 1/3 as well. Yes, the decimal for 1/3 is not the rational. That's rational because it can be expressed as 1 over three. It's rationally repeating 0.333 and so on. It's repeating. We know what's on trees forever and ever. Oilers's number very good. Um the Oilers Oiler isn't spelled like that though. It's spelled like this. Um Oilers's numbers like one that's a 2.7 or 2.8 or something like that going on and on. SS if you don't understand what people saying with SS those are the square root of numbers. You can't find the exact square root for like roo<unk>2 roo<unk>3 roo<unk>35 of course not roo<unk> 16 because that is four which is which is totally why my four coming out like that which is totally a natural number right golden ratio all of that. So you should have un you should have an understanding of sequences of numbers that are recognizable. So um a factor is not necessarily sequence but it's in it's inside this table here factor and multiples multiples is a sequence factors these are smaller factors divider these are exact divisor divider divisor of a particular number a number that divides the given number without leaving a remainder yeah all of this on the Google drive later so the factors of 12 are 1 2 3 4 5 6 multiple equals 5 5 10 15 20 going up the road like that square numbers everybody knows their square numbers anybody can tell me what is 14 squared type it in the chat so I'm not going too deep into this everybody should know what even numbers odd numbers is notation we say even numbers is 2 n for odd numbers is 2 n + 1 so for any value of n 1 2 3 4 5 6 7 2 n will give you an even number where n is a natural number. Odd numbers will be the even number plus one or 2 n and minus one. Prime numbers numbers very important in certain fields of mathematics. Prime numbers are numbers that only have two factors can only be divided by one and itself. Um so used in factorization and stuff like that. So a number with only two factors that can not be divided by other factors besides one and itself is called a prime number. Can you give me an example of a prime number between 20 and 30? Give me an example of a prime number between 20 and 30. Jonathan, between 20 and 30. Are you too early in the morning to start this? Hey, between 20 and 30. All right. Give me an example of a prime number between 90 and 100. This one is asking the multiple choice. was a 99 is not a prime number. 91 is not a prime number. 93 is not a prime number. That earlier that's the whole three things is actually the multiple choice. They'll have 91 93 99 97. Which one of these is the prime number? 97. Seven could go. Three could go into this. Three could go into this. Stay locked in, folks. Stay locked in. Um, and for the one in 20, 23 is a prime number. 29 is also a prime number. Anything else they write is craziness. So compute um here we have the laws of indices. So we we sele we segueing right into the laws of indices. Everybody knows all their laws of indices and understands how to use them. Right out of 10 how much you understand the things on the side here. Any number to the power of one is itself. So if I say a triangle to the power one it's a triangle. Any number to the power of 0 4 to the^ of 0 is one. Anything you raised to the power 0 is one. And there's a beautiful little description that we use to kind of show how this if I have two to the^ five and you two the^ four. This is like saying two by itself five times. Why in the chat if you understand that this is like dividing now by two by itself four times and then think of this we can just cancel and what we going to end up with one two. So 2 to the^ 5 divided by 2 to the^ 4. It comes like you're cancelling or subtracting the powers. So this is 2 to the^ of one which is two. Y'all ringing a bell? Or what about if we decided to have 2 to the power of three and we divided by next two to the power of three. Well then that's 2 * 2 * 2 2 * 2 * 2. Everything cancels. You just end up with what? 1. But that's saying 2 ^ 3 - 3 is 1. 3 - 3 is 0. 2 to the^ of 0 is one. That's why any number raised to the power of 0 is one. Have you seen that before? Let me just um you all just soak down in this for one second. Anytime you see me pause is because somebody messaged me with a frantic message to try and join just make a payment. Why? Why? Yeah. So any number raised to the power 0 is one. So it feels weird but that's how it that's how the cookie crumbles. So we actually just covered a few in the example here like multiplying the power mean add divided means subtract power to a power that's like if you take the whole thing and and and multiply it by itself. Now let me show you what I mean. So like if I have two to the power two and I say cube that that's like two * two cuz that's 2 to the^ of two but then I have to multiply that by itself three times. So how much twos I end up with here in the end I end up with six. So that's why when you say a power to a power it means multiply. Everybody understanding these rules now? Is it finally coming home? Well, second to last day before examin time for you to come home. Um if you have something like this 2 over three and you decide to square both of them, you can square them individually. That's what these two rules here are saying. So you can say that's four over 9. is the same as 2 over three by two over three because you could obviously see that this will multiply this multiplying similarly like if I have x y and I decide to square it the whole thing then that's x^2 y^2 and the last one I'm seeing here is if I have a number and it's a negative power to get the power positive I have to do the reciprocal all right um any admin teacher here with me right now admin or staff I really don't want anybody to miss the crash course even if they pay last minute so cuz takes these back seats after this exam evaluate numerical expression using any of the four basic operations so what is happening here you're noticing that we pointing out each objective in the syllabus so It's like it's like a topical crash course. So you need to know to add, multiply, subtract, and divide. If you don't know how to do that, you're cooked. So the the mass is an AC acronym brackets of division, multiplication, addition, and subtraction. Now, it's not really nice for a teacher to tell you, you don't know how to do it, take it back. But oh gosh, I need to know to multiply and divide and think for five. CXE. So, but this tells you the order in which each one of each one of this goes. Um, and you all should notice now one change here is that this word is usually or order. So, I don't know if Nolini is here. This is one change that that we do. This word is supposed to be order. In primary school, they tell us of to help us learn of, but it's it's order in in every other place in mathematics. Order is another word for what? Order is another word for what? class power. Yes. And power is another word for exponent. All of these mean the same thing or raised to a power. So it means that that so therefore if I have something like um 3^ 2 + 6 to b ^ 2 - 5 [Music] um divided by two. The most strong thing here is what? The brackets. It means I got to do what's in the brackets first. Everybody understand? So I need to come and focus my attention on the brackets. Do you all understand? 32 is 9 + 6. That then what's the next most strong thing? The squared. So after I deal with the brackets and I get 15 to be squared, I have to do that 15 square 225. Then what's the next strongest thing? the division five divided by two. So I have to say five over two and then minus it from that. Five over two is 2.5 and subtract and you get 20 2 um 22.5. This is a little fractions family here that is pretty useful for primary school and it still remains a little useful for uh secondary school especially in the multiple choice especially this family right here. So you all can take a little look at this and remember your good old primary school days and see if you just freshen up on this. This document is gold. If I had this document, I would have taught mathematics when I was in um in 2005. It's giving you everything that you need to know, right? So list the set of factors and multiples of a given integer. So here we go through factors and multiples. Again, I mentioned this already. So for time sake, you'll see me skipping out some of the easy stuff. You want to um read that. I don't think anybody any everybody here understands factors and multiples right is there anybody who doesn't know factors and multiple I don't need to waste time on that right that's like form one compute the ACF socf LCM who can tell me where LCM is used LCM is used in an important thing in in fraction in in especially algebraic fractions so like if I have 3 over 2x + uh 5 over let's say x^2. We know what to do there. How many of youall know what to do there? Why is it you know what to do n if you don't know what to do? See some y some n. So you'll be able to find the LCM of these two things. So the LCM of these two things would be like maybe x² and uh two. How many times will 2x go into 2x 2? x * 2x^2 / 2x bang bang one of the x remains so you get the x multiplying here so you get 3x plus x^2 into this will be two so you get two multiplying here so you get 10 so usually how this comes is a little simpler like maybe 4 over x - 3 over x + 1 and we end up multiplying like this then this one goes over there. So you get 4x + 1 and 3x. So that flow, that flow, look it up. LCM, LCM of the denominator. That's what I'm talking about. It's very popular in question one and question two. It's called, this one here is called algebraic fractions. And if you doing your past papers, you would have seen countless algebraic fractions. How many of y'all do like five pass papers? How many past papers you all do? Paper two. I'm talking about how many past papers have you all done so far? 158 1 3 0 N yeah you're trolling can't be zero you're not serious if you do zero pass papers state the value of a digit of a numeral in a given base so this is like a quick segment I taught this in a whole one and a half hour class I'm about to teach it in like 10 minutes so pay attention numbers can be count we can count numbers in different ways like the civilization I think it's the Sumerian Sumerian they didn't use base 10 most civilizations use base 10 we use base 10 1 2 3 4 5 6 7 8 9 10 and then start over 11 12 13 they use base 60 I've been told so uh and some other places like use base 60 and base um 12 and these other bases base two or whatever so computers use base 2 so like to count a computer will say 0 1 and two is 10 or 1 zero and then 11 then 1 0 0 again then 1 0 1 then 1 1 0 1 1 so I went through the whole idea of how like the the first principle idea of this but what you need to take away for Cxc is yeah yes it's out earlier what you need to take away from this is how to convert so you don't necessarily need to know to count in base two but you need to know to take a base 10 number and change a base two number and vice versa. So watch how it's done. Let's say I want to change the number 14 to base two. Well, divide by two and record the remainder on the right hand side. Two goes into 14 seven times. Remainder how much? Zero. Why is in the chat if you understand that first step? Then two goes into seven how much times? Three. Three twos are six. Three fours are eight. So only three pairs of twos I can get here. But there's a remainder of one. Two goes into three one time, remainder one. Two goes into one no times, but there's still a remainder one. Read it going this way. 1 one 1 0 base 2. That's the answer. This is the answer. If you have to convert 12 to base to base two, uh you have to know how to do this ladies and gentlemen. All the time base 2 came, it's been a question just like this, just like this for two marks or one mark. and you had to convert I think it was the number 12 and the number seven or six. How many of you all saw it in the past paper that you did already? So make sure you can do it this way. If you want to go backwards, if you want to go backwards, you'll use this idea. So let's take a random base um base two number. Let's say I had this number one 01 01. I think that's 20 something, right? So let's say you have this number and I want to convert it into um this is base 2. I want to convert it back to base 10 or da or decimal system. Well, you put one 0 1 01 and you have to know the place values for the base. To get the place values for the base, start with one. There's always a ones column, but instead of going to 10, which is base 10, you'll go to the two cuz in base 2. Then say 2 * 2 is 4. 4 * 2 is 8. 8 * 2 is 16. Everybody catch that. If you was using base 3, you'd say one. You always start at one. Go to the base. 3 * 3 is 9 and not 6. 9. 3 are 27. 27 * 3 is 81. So that will be a place values base. M let it soak in my brother. So this is 2 to the^ 0 2 the^ 1 2 cube and 2 to the^ 4. and so on. That's how we get place values in normal system. Jacqueline, have you ever stopped to consider that the place values are 10 squared 10 to a power place first place value is one that's 10 to the^ 0 next place value is 10 10 the^ 1 next place value is hundreds 10 squared next place value is thousands 10 cube so you use the base and you raise it to the power for each place. So what are we going to do with those numbers? Well, you multiply. So you multiply down like this. So you have a 1 by 1, 2 by 0, 4 by 1, 8 by 0, 16 by 1, and then you add. So 16 + 4 + 1 is 21. That's the number. As simple as that. As simple as that. So that's how that's how you learn how to do this. So we just converted this number to base 10 and we figured out that it's 24. You can see the the working here. These are the place values. So maybe here we could insert the word place values. Moving right along. Convert from one set of units to the other. Everybody should know how to do this. Excuse me. So the strongest way to solve for this just now. Let me take a little scary little peek over on the chat just to make sure people are getting through the last minute. Anybody here did this organized yourself for this really? If you didn't, I want to congratulate you. Well done. Um we are a little overwhelmed with the amount of students um who just register in the last two days. So it's hard. I I can't just you know magically convert from one set of units to another. So the idea with this is when you're converting, you need to be able to find for one. It's a primary school idea. Let me show you what I mean. So like if I said one apple is $5. How much is 10 apples, people? 10 apples will be $50. Why do we know this? Once you found for one, you can solve. There is going to be something in the exam in question one where you have to find for one 99% sure. So what is finding for one? Finding for one is the step before this. So what about if before this I told you I paid for five apples I paid $25. What is the price of 10 apples? Well, there's a ratio here. I could just multiply by two. But let's say I couldn't find the ratio. You'd find for one first by dividing. So it's important to understand finding for one is a division process. So if five apples is 25, one apple is 25 / five which is five and then you go forward. Why is in the chat if you understand that? So it will show up in measurement as well. Like if I said, "Okay, how much is um 7 cm in millimeters?" Well, find for one, find the conversion. 1 cm is 10 mm. So 7 cm is 70 mm. How about if it's vice versa? If I said how much is Let's do it in a kind of reverse idea. I have 400,000 cm and I want to know how much kilometers that is. What I could see quickly is 4 km. 1 kilometer is 1,000 mters. Yes, there's a typo here. Thanks for that. Everybody look out there. There's a typo here. Uh 400,000 cm. So, watch out for that. When you see the note said, don't go and think 1 km is 100 m. Um so, I have 400,000 cm. I change it to kilometers. The answer is not 400 by the way. It's four. But how do I know it's four? Well, I could do a fine for one idea here as well. You know, I could say, well, what's the conversion? I know I know this. 1 km is 100,000. So, let's write it in reverse. 100,000 cm is 1 kilometer. Find for one. 1 cm is 1 divided by 100,000. So, now I have for one. If I have for one, I could find for anything. So 1 cm is actually one over 100,000 kilometers. A very small piece of a kilometer. So therefore, how much do you think is? Wow. How did you reach down quite there? Zach, now you controlling dragging 400 400,000 cm 100,000 multiply by 400,000. So cross cross and you get four kilometers. So this is the finding for one approach very strong very powerful um method for you to solve um any questioning units and no sorry the other thing to to remember is um the area and linear scale. I spoke about this at length a few times. I even mentioned it in the last crash course. The area scale and the linear scale, it shows up in stuff like enlargement. And the idea that I brought forward was something like this. If I had a square and I told you one side of the square is 3 cm and the next side of the square is 3 cm, how much is the size of this square? How much is the area of that square? The area of that square is going to be what class not not this one square here you know the whole square was the area of the whole square so one side is 3 cm the next side is 3 cm was the area of the entire square so yes it's 9 cm squared now if I told you I raised this by from 3 to 6 cm what is the scale factor here The scale factor is what? That's called the linear scale factor. And all scale factors given without context are always linear. Scale factor is two. So 3 to six it raised by two. Multiply by two. You can't multiply the area by two. So this is 9 by two. This isn't 18. Is this 18 cm squared class? No. 6x6 is actually 36 cm squared. So what went on here? Well, it turns out that the area scale is the square of the linear scale. So I actually have to multiply by two squared. If I want the area scale factor, I have to multiply by 2, which is 4. 9 by 4 is 36. You catch that class. Alia, what are you talking about? This is six. So it's 6 by 6. So let's just see real quick with another idea here. Let's say this was two and let's say this this size is 10. What is the scale factor here? Five. So the scale factor is five. So what what would be the scale factor for the area here? The area scale factor 5 squ which is 25. You actually multiply by 25. Let me show you. The area of this is 2x2 is 4. The area of this is 10 by 10 which is 100. If you had say the area of this is 4x 520, you're getting it wrong because 10 by 10 is 100. So that idea is very important to keep locked in that conversion of squares or cubes. You have to pay very attent uh keen attention to what's happening there. Close attention. So here we have a couple conversions that we see pop up all the time. * 1 m squar is 10,000 cm squared, not 100. 100 * 100 is 10,000. 1 kilm squared is 100,000 squared cm squared. A very big number. One liter is 1,000 cm cq. So pay attention to that. You should know to convert from 12 hour to 24our clock. Is there anyone in class who doesn't know how to convert from 12hour clock to 24our clock? Speak now or forever hold your peace. Everybody know how to do that? So add 12, right? Is as simple as adding 12 or subtracting 12. If you add 12 to this, it goes to 1630. There's some dots that we normally put in there. Hold on. Then we go into significant figures, decimal places. Significant figures, decimal places. Approximating to significant figures and decimal places is something you should know how to do. Like if I say 17.352 to one significant figure, what is the answer? What is the answer? If I want to go to one significant figure G, if I want to go to one significant figure. Wow. Wow. Nobody gets Oh gosh. large strength right to is the first person to get correct to one significant figure this is 20 what going on boy what going on here so one significant figure this is 20 class one significant figure mean you start from the first significant figure keeps its place value you know it doesn't change magically to one it's you round up Yeah. So you watch at this you look five or more add one. A lot of you always giving me one decimal place. This is different. Wake up some. You need to go and take a run around the house and come back. Do some more than on your face and come back because what is this? This is a very easy question, right? Um if I want to go to if I want to go to two significant figures, what's the answer? Two significant figures is 17. Very good. If I want to go to three significant figures, what's the answer? 17.4. 1 2 3. Check what come next? Five or more. Add one. So 17.4. 17.4 is also the answer. If I wanted to look at one decimal, please. So the first decimal, please. Catch it. Um, okay. So here's some examples for you to look at. Right? This is one significant, two significant, three. Make sure you understand why as you're reviewing. So your eyes should just pan through this and make sure you understand all of them. Don't make no silly mistakes in the exam. Decimal places is different. So decimal places start after the decimal point. First decimal place, second decimal place. So let's say I had 47 uh 326 and I want to go to one decimal place. What's the answer? One decimal place here would be what? Type it in the chat quickly, folks. 400. Oh, I seen I just take a peek on on on YouTube here and I see Rachel. Rachel, you get the admin yet. message the admin your receipt or whatever so they can add you to the group so you can use the meet cuz I didn't know Rachel is actually questions so you'll be able to ask your questions um so 47.3 is correct not 4 I hope nobody have point 2 Jacqueline why you rounding we don't ever do that we don't do that here rounding down is not part of approximation if we round down it means it stays the same understand If I want to go to two decimal place, what will be the answer? Two decimal places 407 33. Very good. Basic stuff, right? Then there's this that comes. Everybody know how to put this thing holding to the calculator. At this stage in the game, you're supposed to have the best calculator in the business, the Casio. Where did my board go? So mean to say my gorgeous cher disappears from here bro. So immediately once you once you see this once you see this put it in your calculator one time even if you're not allowed to use your calculator for the question you'll still um it's not that you're not allowed to use your calculator if they want you to show the working you still want to get the answer so you know what you're working towards immediately this might be the first question you see in the exam immediate calculator you're writing this one 5 over six. Um, did they switch my fraction? Why not see my whole fraction? Yeah. 56 equal. Now, if they said they want the exact value, what are you supposed to do? If they want the exact value, leave it as a fraction. You could even leave it as an improper fraction. is not a not a problem. You could even leave it as an improper improper fractions are totally cool in mathematicians. Um so 36 over 35 is the answer. And if we have to go to decimal places, what do we do? Well, then you press the equal, you get that little S. You pay 1.028 and you check how much decimal places. Three. If we have three decimal places, 1 2 3 5 comes next so raises to 9. Three significant figures 1 2 3 8 comes next. So we raise that to three. Very good. So these are your answers here. 36 over 35 1.029 1.03. So you you are going to get some kind of significant figures, decimal, place, whole number, approximation, some fraction, something in the first part. They've been doing that for the for the better part of three decades. You are going to be fed some food here. Don't lose any silly marks. Some ideas now to take away that Miss Nolini will get a teacher. I cannot believe I let Miss Nolini get away with the algebra next year. Next year additive identity is zero. Adding zero to a number doesn't change it. Remember we spoke about that. So if I have a plus 0 is the same as 0 + a is a. So zero is called the additive identity. the multiplicative identity is one. So in math when we hear the word identity means doesn't change it doesn't change. So if I have four * 1 the number is four. It remains the same as the original input. So that means one is the multiplicative identity. Using the identity and the operation the uh the output doesn't change. Additive inverse the inverse creates the identity. So if I have four and I add its inverse here, I will get the additive identity. So 4 minus 4, the inverse of addition is subtraction, it created the additive identity. And this idea transfers over very neatly into matrices. You know, um same thing like if I multiply 3 * 1 over 3, I will create one, which is the multiplicative identity. Therefore division this fraction is in the form of a division that's three one divided by three and this one multiplies by whatever there um it creates the uh addit multiplicative identity. So the inverse of multiplication is division and most students can tell you inverse but sometimes they forget what identity means. There are three laws that show up in the exam. The associative law not so much. The distributive law this always shows up in terms of expansion. And the commutative law is an idea that you need to am I sticking people? Is my screen sticking for you? You hear me clean? Great. Great. Great. Right. So once it not sticking for some people that mean it not sticking or if it's sticking for you that mean that you issue that could be the amount of people on right. So just saying those of you all who on the Google meet if the Google meet is too much for your device you can always use the YouTube link it was sent in the group. Um uh where is it? Where is it? Let me just send it back over. at the last minute in in bites for okay so what I'm saying excuse me what I'm saying is this commutative property is tested who can tell me what topic this shows up in it shows up in a couple of topics I forget the next one oh yeah which topic does it show up algebra okay but no like where it's tested as a thing like is this commutative shows up in algebra but specifically what in algebra or there's another topic yes matrices I mean studying it shows up in matrices they will ask you if a b is equal to b a commutative is matrix multiplication commutative or in what specific topic in algebra does it show Sometimes it shows up in not. Yeah, it could show up in functions, Jonathan. It could show up in functions. That one a little rare. So like maybe they could ask if fg of x is the same as g f of x or stuff like that. Binary operations TN is so smart. That's correct. So like a star B star E they will ask if it's commutative and that's what you have to test the fancy um substitution very good. So commutative just means it works vice versa. So 2 * 3 is the same as 3 * 2 order is not important. So read all these things is very neatly written here just for quick memorization. Um so like if I have 2 minus 3 that's that's -1 3 minus 2 is one. So subtraction is not commutative division is not commutative addition and multiplication is commutative and then sometimes you can get a specific binary operator which molini will so associative law and distributive law so associative law just speaks about the order in which we do the the the grouping. So like if I have 2 + 3 + 5 2 + 3 + 5 the normal person will say 2 and 3 is 5 5 and 5 is 10 right anybody in the normal crew here but it's also the same as if I say 3 + 5 is 8 un the abnormal group so 2 + 8 is also 10 so grouping it doesn't matter which one I group and do first that's associative law it it obeys the associative law of course subtraction and division doesn't obey the associative law only addition and multiplication distribution speaks about how one law interacts with one operation interacts with another operation. So does the operation distribute across addition, subtraction, whatever. So for example, if I have multiplication, I can try to see if multiplication will distribute across addition. And of course, we know it will. Here's what I add distribute across addition means. 3x 4 is 12. 3x 1 is 3. Answer is 15. 4 + 1 is 5. 3x 5 is 15. So since these two are equal to each other, it obeys the distributive law. Everybody understand? This is three multiply. I just put the times to show you that it's multiplication going on here. Okay. So that's that. That's the laws scientific notation. I can't believe he's still on scientific notation. What happening here? Page 19 out of 296. Progress is progress. Yes. Time 10 to the^ B. So, everybody should know how to do um scientific notation. If I have 401 and I want to express it in scientific notation, that's 4.01 * 10 to the power of how much? What number do I put here? Two. because I would have brought the decimal place two places to the left. Now moving to the left is a negative i there. So really what is happening is by saying it's positive2 I'm actually moving to the right. I have to multiply this decimal place by 100 10^ squ which would means it will move to the right. So a lot of people really confused by this idea in standard not scientific notation. I have 0.0046 and let's say I want to express this in scientific notation. The answer is going to be time 10 ^ what what is A and what is B? What goes in the place of A? A would be what? 4.6. Very good. And what go in the place of B? Negative3. Very good. Now some people say well you move to the right that's because on a number line if you go right that's positive direction. So why do you put negative? Well you yes you carried it over to the right but really this 4.6 is being multiplied by some number. If that's positive if that was 1,000 that make it a big number. You understand? It's in reverse. You need to think about it. So actually you're not you're not moving it to the right per se. when this turns -3, you'll have to carry this back across to the left. So, it's just the way we think about it. But for if you want to just memorize it like okay, move to the right is negative. Move to the left is positive. Just for scientific notation, that's perfectly fine. So, there is the tip and here's some numbers to express in scientific notation. I'm not going to show you the answer. I'm going to scroll past the answer. You can do those for yourself there. Calculate any fraction or percentage of a given quantity. Express one quantity as a fraction or percentage. Compare quantity. So this going into percentages. What is the one thing in percentages? The one trick in percentage increase that you have to look out for. Everybody please if you come for the exam, no one will lose any marks from my class. Percentages do not attack the final value. Doesn't attack the final value. Right? Enough your fall enough holes with that. Here's a little table. It's in the notes as well for you to just get that in order. Now, the table could look a little confusing. So, I'm going to show you some examples. It's saying that if you have the initial value and the final value, get the difference where the initial multiply by 100. That would be how we would find the percentage. So, in other words, this is when you want to find the percentage increase or decrease. This is when you need to find the final value. And this is when you want to find the original value. This is the bad boy that you got to watch out for. Everybody and their grandmother and the dog and everybody know to do one and two. That's that's bread and butter. That's the first thing you're doing. Percentages all the time. Everybody know to do that. It's the last one that everyone makes the mistake on. So like if I told you I had $10 and now I have $14. What is my percentage increase? Immediately everyone know what to do. This is your initial value. This is your final value. So you say 14 minus 10 that's four. You understand the difference. And you take that difference and you put it over the original. Always over the original. Multiply 100 over one. And you get a percentage. Why is in the chat if you understand that? That was your method is going to use right. That's totally fine. Everybody could find the percentage if you give it the initial and final. Once again, if I just give you the initial and the percentage, even better. Everybody loves to do this one. Like if I said I had $200 and I want to raise this or let's say I want to drop it by 25%. 25%. So everybody know even people you might even have fancy methods instead of multiplying by 25%. Some of you all might multiply by 75%. So you do your 0.75 of this and you get the answer one time. Right? But the long way might be you say 25 over 100 multiply by 200 you get your 50. Then you say 200 minus 50 is 150 and that's your answer. Why is in the chat if you all understand that? So those are the two easy ways. What about if I told you I have um $400? This represents an increase of 33 and a3%. Let's not even make it so complicated. This represents an increase of 10%. What was the original value? Type it in the chat. Type it in the chat. Make sure you know to do it. So, I I I reached up to $400. This is my final value. I'm trying to find how much this was. The original value I increased by 10%. Type your answer in the chat, folks. I just want to see answers. Me want to see working. First time I asking for answers are no working. type of answer. So my final value is 400. Anybody have a different answer? Cuz like I feel a pain in my chest. Oh gosh. All right. So people look at the 363 point wherever it is thank thank for for listening to me. What is the most important thing to remember with percentage increase? If they give you a final value, what you need to tell yourself immediately in the exam, the reason there's a concerned expression in my face is the chances of this coming. I hope it doesn't come because real people still fall in f is show up real plenty. Percentages do not attack final values. You cannot find 10% of the 400. This 10% cannot apply to the 400. There's one way to do it. If they give you the final value, that's this example. And they ask you to get the initial value. Don't jump down and say 10% of 400 is 40 minus 360. Everybody who get 360, no wrong. Zero. No max. Not even a mark. Not even a half mark. I want y'all to feel the pain of getting that wrong. Like are you getting register for some of you all yet? You're still while jumping in to that. The percentage cannot attack the final value. You have to do this. You have to say Maya, you know, you know how much time I say this 100 next year going and stand back on new class and I say it 100 times. 400 110% represents 400. 1% represents 400 over 110. The 100% is 400 over 110 multiply by 100 and you'll get your that answer that 363 whatever point whatever that everybody else getting 64. In the other crash course I showed the new students this as well. I'm going to show you all it again. I'm going to show you all why it doesn't happen. So maybe some of you are like nah that wrong and I I find my 10%. No, you will get no marks in the exam. Math teachers love to ask this. I need you all to know maths teachers love to see a lose marks in this question. That's why only showing up all the time. So let me give you a clean example and watch it. Right? 800 is my original value everybody. If I get increase this by 10%. How much will I get? How much will it go up to? Use your calculator. It's the original value. Don't be afraid. You can attack original values with the 10%. I will get $80. My it raised by $80 and I end up on 880. Everybody understands that this is my initial or like my cost price or my first value. This is my final. Now, pretend that I didn't know my initial value. Pretend that I didn't know that it's $80 it rose by. Pretend that I just told you it's 10%. I increase by 10% and I end up on 880. What some of y'all are doing is saying, "Oh, 10%. Let's just go run down this $880 with this 10%. Bang bang bang bang. Get my $88, which we already know is wrong because it's supposed to be how much?" 80. So you It's It's illegal. That's that's why I keep saying percentages do not attack final values. This is you cannot find 10% of 880 if 880 doesn't represent the 100%. 880 represents how much? This is the 100%. I added on 10% or subtracted depending on whatever I'm doing. 110% or it would have been 90 if I was decreasing. So I have to say that 110% represents 880. 1% would represent 880 divided by 110. 100% would represent whatever that is multiplied back by 100. And of course that gives me the correct $800. If you understand right, are you in the chat? Never make that mistake again. If it comes in the exam, it's come real subtle. It'll come in question one and no big boy question come in question one or question two. Very subtle. And every time they bring percentage increase, this thing showing up order set of real values. I'm not going to spend any set of time on this. You're supposed to be able to put numbers in order. Um if you have numbers that you can't find there, uh like root two, you're not seeing what it really is. Just put it in your calculator and get the decimal equivalent and then order it. Order it up. Compute terms of a sequence given a rule. So, we're going to spend some time on this. What question does this objective show up in? It's an early objective, but what question does it show up in? Investigation series and sequences. So, this is a series and sequences. Question seven. Very good, Angelica Jonathan. Um so like if I have a sequence 2 4 6 8 10 the rule will be 2 n but firstly you need to understand what the n stands for. So the n is like the number the figure number or the term the nth term. Let me show you what I mean. So this column here n stands for my my my terms and this will always be natural numbers 1 2 3 4 the first term second term third term fourth term and so on. Now if I decide to put a se a sequence here no not shia this is the past paper crash course was April this is a topical um topical crash course however if you notice I'm every topic that I'm doing I'm showing you exactly what will be tested on in the exam so useful very useful still so um let's say this is the terms this is the this is whatever it could they could have all kind of things here like the sides of a bench and all kind of different um variables to go here. And let's say the numbers were three, five, 7, 9. How do we get the expression for the nth term? Well, very good, Zach. I see you remember, and Jonathan, as well, this is what you have to do, class. Check the difference. That's the first thing you want to do in the exam. Check the difference and see if you've seen a common difference. If it's like two, praise the Lord because if the difference isn't common, that means it's a scary one. So two and you'll put two n. So this is like 2 n. But if I multiply one by two, I will get two. 2x two 4 3x two 5 4x two 3x2 6 4x 2 8. So what you notice? What you notice about all those numbers? What you notice about all those numbers? They short by one. I just need to add one to reach up. So that's why I'll add one to this. So this is a very very powerful idea for solving series and sequences. Most of you would have seen me do it already. If you haven't, please go look at series and sequences on YouTube curing spring and you'll get a longer view on that. At the moment we are speeding. So this sequence will be 2 n plus zero. Just 2n because there's no need to balance off anything. Yes, this is the first one. Second, third, fourth, fifth. 5 by two is 10. 4 by two is eight. So very nice. First, second, third, fourth. Watch how much it's going up by. It's going up by three. So this is 3 n, but 3x two is 3 by 1 is three. 3x 2 is 6. 3x 3 is 9. All of them are short by two. So that's why we decide to add two. Yes. Very meta. very um comes all the time. Comes all the time. If it doesn't come like this, that mean they hate on it. This can show up in a in question seven. Here's like an example. The I'm going to take a peep at my YouTube people. YouTube YouTube people say, "Hey y me close a couple things on this computer." Right now my computer is struggling and I know the main problem is garage band and all kind of thing open here logic pro computer bleeding right now. So to draw a figure four, this should be easy to do. Everybody could draw a figure four here. Well, you just it'll be this figure and you just have to add on a little extra piece like that. Fill in the mason space in the table below. So you'll see stuff like like this. And usually like every single time that it has come in the exam, I could figure out the table without even looking at the previous stuff. So 1 2 3. Okay, fine. It's it's n cuz this is the exact same thing. 4 7 10. Um, what goes here? People, let me don't rub this rub while you're doing this one. It's going up by three. So, it's 3 n and 3 by 1 is three. 3x2 is six. Short one. 3x 3 is nine. So, I need to add on one. Sometimes you might need to subtract. So, divide a quantity to a given ratio. So, ratio could come in a couple different ways. Those of youall who remember when he did ratio last year July second class or third class or some kind of thing anybody you could remember that far back I said the important thing for ratio is to find for one find for one usually finding for one would have you first finding the value of all the parts or if not finding it identifying the value of all the parts sometimes it's not all the parts sometimes it's just a specific number like three parts might equal to $10 and you're fine for one part fine for one is the key. So suppose that Mr. Kim shared $60 between three children, John, Cle and and Mark in the ratio 235. Here's how we'll find for one. We have a total. So we we identify the total. How many parts represent the total? Total number of parts is 2 3 + 5 is 10 parts. So we're going to say 10 parts represent $60. Very good, General. So one part will represent $6. So then we can find each individual. Once you're fine for one, the party done. Fine for one, party done. Uh, two parts would be two * six. Three parts will be three * 6. Five parts five time six. And you individually figure out how much each people get. Rate out to 10. How much you understand [Music] that? Here's another example. Kate and R share a bag of chocolates two to five. Rare received 15 chocolates more than Kate. So, do we have a total in this example? Do we have a total in it? No, we don't have a total. But can we still find for one? Yes. 15 more. There's 15 more. This is two to five. So, how much parts represent 15? 5 minus two parts is three parts. So, that three parts is the more the three parts represent the 15 chocolate. So, this is the total number of parts. Seven parts. But it's not useful here. Seven doesn't represent 15. Five minus two. The difference in the parts is three. Three parts represent 15. And then you're fine for one. Once you get a number of parts representing a quantity, you get fine for one. One part is five chocolates. And then whatever what how many chocolates? Seven parts. Seven fives are 35. Class. We good with ratio? Ratio is usually food. You need to solve problems involving concepts in number theory. So ratios, ratios and proportion. Finding for one is is a division process. To find what you want, you multiply. So proportion, another word for proportion is what class? Another word for proportion is my wife just dropped off some um food here. I've seen grapes. I've seen lettuce. bread, eggs, an apple, a vitamin. Um, another word for proportion is not part. Another word for this proportion topic is is variation. Dearation now in English proportion mean a part. Yes, but this is the topic. Variation is what they're talking about or proportionality. So ratios, rates, and variation. So let's go see what some of these things entail. Oh, by the way, what is the key thing with variation? The key for variation is to find what you find K. And what do we use to find K? I think I have it further down. I think Miss Nolini will do that in algebra. So anything miss do, I not do it. Calculate the discount, sales, tax, profit, loss. Oh my gosh. consuming arithmetic. Somebody say section two. Somebody say one minute break. I know it have a timer. I think I can do here. Somebody say 2 minute break. So 2 minute break. I go and use the bathroom. It counting at the top of my screen here. Why using the bathroom? I eating and reading comments. 2 minute break. [Music] Come on. Where's the question? Send my dinner. Uh yeah, we're doing every single thing on the max. I don't know what you're talking about. Yeah, I'm trying to finish my I'm so that was that was bread and I think I will make bread. an egg, bam, grapes, all kind of thing. Yeah, Kiana. So, while doing the YouTube link, it will stay there so you can watch it immediately afterwards. So, discount, sale, tax, profit, loss. How many of youall know to do that? The timer I go on market done with the time. So you all know to do your discount sales tax profit and loss. Nice. So discount is a reduction of the mark price. Opposite of discount is a mark about that discount mark price selling price. So there's a difference between the mark price and the selling price. Right? So, if there's a t-shirt on sale and on the tag mark $100, but then you eventually pay $90 for it because there's a discount of $10. The selling price is $90. You all understand what I'm saying? The selling price is $90. The math price is 100. So, you might be saying, where cost price come? So in maths we normally talk about cost price as the price that the seller pays for the item. Like let's say he bought the t-shirt for $80. That's his cost price. You all understand? He bought it for 80, try to sell it for 100. You come here say the best price and he say all right take it for 90. So then we can start to talk things like profit loss profit percent and stuff like that. Right? So profit financial gain loss no financial gain decrease of finances in the transaction. Easy basic stuff. Profit is the selling minus cost price. Loss is the cost price minus the selling price. You can calculate the percentage by putting the profit over the cost price. Do you ever put anything over the selling price? Do you ever put anything over the selling price? No. No, you don't do that. Selling price is a final value. That's that's the breaking the low again. You're trying to do a percentage of of the final value, right? So the percentage loss is the loss over the cost price. Again, very important tips. Let's see what's important about it. Original value talk here. And which one to subtract, right? It's subtracted from the marked price, not the selling price in case they give you a bunch of different prices. some problems involving MA price, selling price. But that's enough talk on M price and selling price, bro. All right. So, higher purchase and mortgages. How do you all feel about higher purchase? Higher purchase is usually a really nice part of maths to understand. It's not too bad. It feels preaching. Anastasia, you have problem with higher purchase. You got to get good at courts and standards and rub. So higher purchase you have your down payment and you have your installments. So basic stuff what what tricks could come in higher purchase? Nothing really. It's just sometimes at the end excuse me for one second. Hello. morning speaking but I am teaching a class right now touching B from social bank tomorrow around the same time tomorrow Nevi um solve problems involving payments by installments as in the case of higher purchase and mortgages. So guys with higher purchase the one trick that could come is is um the year like if you get a year like 14 the answer 14.2 two years. What are you supposed to do? 15, not 14. General 15. Even though rounding off right now supposed to give you 14. Yeah. Always have to round up with these cases. So, this is one of these situations where you can end up rounding up like when you're working on the um time. So, higher purchase is really the down payments plus the installments. Like, let's say there's this TV on sale for $5,000. You can't pay the 5,000 one time. So they tell you no scene pay $1,000 now and then pay five months or let's say 6 months of $800 each. So the value of the installment is 800. The total value of all the installments will be 6 * 800 which will be how much? 68 are 4800. And this is all the installment, but don't forget you had a down payment. So, how much you end up paying for your total higher purchase in this scenario? You'll end up paying the 1,000 down payment plus the 4,800, you'll end up paying $5,800. So, this 5,800 is your higher purchase price, which includes your down payment. Sometimes there's no down payment plus the installments. Usually the higher purchases higher than the what we call this what we call this value here is CP, but it's not cost price. It stands for what? This is the what's that word? Cash cash price. So the cash price is the price that if you pay one time. Cash price is the price if you pay one time. Just know I just see a couple other frantics. N no no no y'all look what this who is this faith who do this faith that you who could tell me what is the first when I see this I getting scared one time what's the problem with this sh people why I don't do this on the They're asking back now. When you do this, you're asking for a shop to teach your M. So, please use brackets. I don't know what it is you're trying to say here. What was this question? 2 minus one and 17 divided by 5 over 6. That's what you're trying to say. Let me tell you what your calculator think. Your calculator say 2 - 1 / 7 / 5 / 6. That's what your calculator think you're doing. Yeah. Confused. So you need to you need to you need to use brackets. So remember this is B mass division and fraction fractions real powerful. It's sticky. So you need to actually say 2 minus put this in brackets. I don't like to use this one over thing in this calculator at all divided by 5 over 6. So try this and you may get better you may get a better result here. Try this situation. Right? Some of you all think you're you're writing this answer in the exam confidently and confidently wrong. I'm sharp people brackets brackets. Oh, this came once. I think it came once a long time ago. Why people use higher purchase plan instead of paying cash? You do not have the initial amount. You don't want to spend the money right now. You can use the item right away. Like I use the example of a a double lender. They can use the this kind of going into POE and one of them business get in business. They can they can take the high purchase plan and use the fridge right away and [Music] then make back the money to pay it. If they didn't buy the item, they can't make no money. So, bill credit is another reason why people use high purchase. Simple interest is usually one of the simplest things that can come in the exam. What are some of the tricks that you need to look out for in simple interest? So the same thing with the time. The same thing where the time could have to round up or sometime they could be asking for the amount and the amount is the extra formula that some people don't know. The amount is what actually there was one really simple example. The amount is the principal plus the simple interest that came and I gave it in class and nobody was finding the simple interest and it come two times within the last 5 years. Let me show you what happened. So be be on the lookout for this. They they seem to love to do this as well. So they might say Zach invested um [Music] $7,000 in an account over four years and [Music] received 20,000 at the end of the period, right? How much is his simple interest? One mark and the whole class get the whole of the car get correct. the whole of the Caribbean struggling to get get that um pawn mark because everybody just trying to see if they could use this formula like why isn't it working why why isn't this formula working your camera come on so what is the important formula that we must use here there's no rate you can't use this formula I have no And this was happening in class 2 last year. So there's a question like this. Of course, that wasn't in the question. How we not you have to use the formula that I just gave you. Compound interest. You have to use this simple formula. Think about this people. If he invested 70,000 and then he got 80,000, what is his interest? 10,000 the amount was given. So watch out for that. You see how even some of you all are struggling to figure out what to do here. This is the amount. This is the principal. So therefore you use this formula where you say amount is equal to the principal plus the interest. So therefore if I want to find the interest, it's just the amount minus the principal. How much you get in the end minus how much you had in the beginning. And of course for this we will see $10,000 pop out if you understand type IU very sneaky and stuck a lot of people and not even like people who struggle with math like really good math students who like what to do in this scenario. Okay. And then the other formula all have simple interest times 100 and then they'll have the other two ident the other two um variables here. So like if I want to find principal is read by time. I'm going to find rate principle by time. Time principal by rate to find the missing pieces. So for example, the second part for this question would be to determine the rate. And we can determine the rate now because we just found the simple interest is 10,000 multiply by 100. We have the principle. We have the time is four years and we've been able to determine the rate. Classic classic. That is the new meta. That's the new style of questions. Um, Timmy blah blah blah at the end of Look at here. Look at here. This is the question. Hey, Timmy. So, this is the question. So, where's the interest class? [Music] 6,000 Nisha we get 14,000 from. So it's as simple as subtracting and sometimes finding the simple interest. And yes, of course, they decided the rate at Josiah Edwards. Timmy was an old student. Timmy was last year. Zach Josiah Edwards Bank. Jo Edwards was a good student. Not last year, year before. Miss you remember Timmy? Some problems involving Timmy. Zack. to me was a nightmare. Solve problems involving compound interest. Compound interest is when you earn interest on both the money you save and the interest you earn. So this is the formula with compound interest. What different about the simple interest formula here other than the obvious formula difference and exponents and things. But what do you notice a key difference that you have to No, the simple interest formula gives you the simple interest. The compound interest formula gives you the amount straight away. Very important. So if they ever ask you what is the interest earned and you use that formula, you get it wrong. You just find the amount. To find the interest, you need to subtract the principal. Okay? So that's your amount formula. This is a simple interest formula. Um this n here is secretly t. And what is t? Time the number of years. So the n here is the number of years. Okay. But we use the symbol n in this case as the exponent. So there are two approaches for compound interest. You could read part by part or you could read formula style. I want everybody to know we on page 37 out of 298. We making progress. So let's look at some questions with compound interest. It should normally shows up in appreciation or [Music] depreciation. Appreciation means value of something increases over time like relationships. Your relationships will appreciate your knowledge. Your knowledge of math supposed to appreciate. Yeah. good relationships like your friendship, you know, after you know this person 15 years, you trust them, you give them your passwords and thing. I don't know. Um, no passwords. All right. Well, I don't know. Trust them to some degree. The appreci appreciation means it increases. Depreciation means it decreases. So, examples of things that appreciate would be gold, land, of course, land. Everybody know everybody here in Trinidad have a strong understanding of land. I don't know if it's the same in other countries but um uncle the uncles and them will be fighting with cutlas and thing over the land because you know land appreciates the value going up gold and antiques fine art things that depreciate is literally everything else consumable goods um cars machinery my laptop anything on my table here depreciating over time Um so this is the formula if you're using appreciation. If you're using depreciation there's a subtle change that positive is going to that addition is going to change to subtraction. In Cxc you can generally get away not using the formula. In some instances you actually have to not use the formula because they want you to do g by a. So in CX the reason why you can get away not using the formula is because n is always three years or less. Usually it's two years. I cannot remember the last time I see three years. Meaning you could just do it step by step. Let me show you what I mean. James bought a car for 150,000 depreciates 10%. How much it worth in two years? So yes we can use the formula. Did I use the formula or not? So we can use the formula amount is principal by 1 plus rate over 100 to the power of n. N is what? What will N be in this question? Two. So you put this. Thank you. I was just checking to see if I was paying attention there. So you use you use a subtraction there. And this would be 150,000 and 1 minus 10. I could I can see this is 0.9 squared. And you put that in your calculator. And how much you're going to get? Whatever you get right slow I need a little calculator crew to help me out a little faster Shane it can never be oh multiply the 0.81 need one by the one, right? So 121,500 be nice have a little space between the 500 get problem to read numbers and things sometime. So cool. So this is the dollars. This is the short way. This is the uh formula method. Everybody cool with that? You just plug it in and you get through. But what about if they answer the first year and then the second year? Then it don't make sense to do this way. do it the short way where you say okay take off 10% of this which will make me end up here 135,000 meaning find 10% of this and subtract it or the shorter way even is to find 90% of this because you take out the 10% of 100 that's what I did here everybody understand that value and then do it again because this is after the first year so you need to depreciate it again find 90% off of this and you get the value that we get with the formula so this is the stepbystep approach which shows up a lot. Oh, Alia, I supposed to sit down after supposed to sit down after 1 hour. 1 hour and 40 minutes gone already. I'm tell you I trying to make my dead here. Peace in peace in for the adm double check on that for me. I don't want to mash up my knees now. You're supposed to give my little I teaching you all the tips and tricks in mathematics. You can just remind me so you know. All right. So this is the formula method which we just use. Solve problems using measures and um money. What I'm talking about here? Okay. Consumer arithmetic. Excuse me. So, one of the is one of the topics that we see this showing up in is exchange rate. If I had to pick the topic I hate the most is exchange rate um in mathematics. Although I love all of these CC mathematics, but the reason I don't like exchange rate is because there's no like shortcut or no tip or no trick or anything. You just have to do a bunch of silly calculations. So the one of the strongest tip is find for one. So put what you are trying to find on the left hand side. This is from the ratio from the exchange rate. Yes, Tiana, you can join just once you reach um which is what most people do. Divide by the 6.75. This is the find for one. Finding for one is a division process and multiply to find whatever you want to is. All right, let's see. It seems like you wanted to find 1350 in this case. So you multiply this and you get the answer. This is a past paper. That number ring a bell. So let me just give you a quick one again. What is the currency anybody? Um Lita is the currency rate for um EC and USD 2.71 or something one USD is equal to how much ECD? 2.71 2.7 2 All right. So 2.72. So let's say I have $400 EC EC dollars. How much is it going to be in US dollars? Somebody work on that real fast. So I mean by eating my mic and mic on YouTube. So everybody in YouTube just here needing chop chop chop chop chop YouTube people are sorry yes it will be available for the person who was asking on YouTube it will be available [Music] um so everybody understand how to do this so the long way if you don't have a method the long way is to put 2.72 EC C wait what I trying to find US ECD is one USD so one EC is equal to one over 2.72 and if I want to find the 400 it'll be 1 over 2.72 by 400 so just finding for one approach will help you if you don't like if you if you struggle to see how to divide these things whether you're dividing or multiplying which is usually most people's um dilemma with this do I divide do I multiply if you use the find for one method uh solve problems involving rates, taxes, utilities, invoices, salaries, and wages. So, let's take a look at all of these. Hold on. Rates and taxes usually tied into salaries and wages. So, which one of these things usually um shows like this? tax on your salary. It will be the salary side. Which one shows over time? It will be the wages side. So, usually these are the two most popular topics here. Why? Because they can add so much things from it. Shopping bill is like the easiest one. And so, if this comes, you know, they love you. Shopping bill is like the easiest thing that can come for consumer arithmetic. has the bill where you have the quantity and you know what I'm talking about uh the item the quantity the cost price for one and then you add it up and then you have the total and the VAT that is literally the easiest thing that could come for for consumer arithmetic I think and utilities these haven't been showing up like I I have not seen it's like they're off the syllabus but it's not so so hi so utilities these utilities is um something that you need to know about. So let's just check some of the tips when calculating total electricity cost or total water cost or any other utilities. The trick with this that used to come back in the days is that there's a variable cost and there's a fixed cost. So usually there's a cost that is the same monthto month like you're renting the meter rental and then there's a variable cost like cost per liters used cost per kilowatt hour. So you need to work out the both of them. There's some tricks also with meter reading. One, you need to be able to understand how to use the zeros here. Like if I add two, if I add 9,000 and all right, let's say I add whatever to this 001 1 0 3 1 0. You understand? So you keep the leading zeros there with the meter enter. Also there's some subtraction ideas. Okay, let's see what's going on. Consider the following information about the electricity bill. Rental fees 100. This is a fixed cost. Every month you have to pay that. This is the variable cost unit per unit. So if you use this amount of kilowatt hours the month and sometimes instead of telling you this directly they will show you. So like I might they might say all right initially his reading was this. This is his reading at the start of the month and at the end of the month his reading is this. So how much kilowatts you think he use? You know that you subtract and in between he use 70 class. So sometimes you have to work out this from the meter reading. So once you have this, you multiply this by this value. That will give you the variable cost and you add on the fixed fee. So you pay $100 plus the 1500 watts kilowatt he used, you multiply that by the 0.065 and you get your pen up this in physics. Recall that meter readings it continues to increase. So if his meter reading is this and he used this one February, ending of February, you add on the th00and this will be his meter reading. Hey, I really love this note boy. Real nice. So, this is a shopping bill idea. 3 by 100. Um, 3 by 100 is 300. How we get the B? What is B going to be? 250. You divide uh 900. What is C going to be? Three. Everybody know how to do this, right? The primary school things. Then you add up all of this. 300, 500, 900. So, how much you get? Um, no. 1,700. You get 1,700 and then you v you work out 18 of this. Uh, somebody how much it's going to be. All right. $212.50 and then you add it. So you get 1 [Music] 91250. You all understand why I said this thing could come and when this come is get one mark one mark one mark uh one mark one mark one mark 1 2 3 4 5 six marks just running into your paper immediately. So good times good times. Here's another example, another um thing from that objective wages. So Ryan works 40 hours at a rate of $10 per hour. Over time is calculated at time and a half. What does that mean? If they say double time, time and a half, you all understand what that means. Time and a half would mean how much is his overtime rate. Then matter of fact, let me go to the questions one time. What's the basic wage? What's the basic wage? Quick, quick, quick. Just give me the answers. No, that's the basic rate. $10 per hour is the basic rate. So, the basic rate means per hour or per unit time or per second. $10. Basic wage. Wage means per week is 400 or per quantity of time. Um, how many hours did he work? Overtime two. What is the overtime rate? So we have the basic rate is $10. The overtime rate will be $15 time and a half. 10 plus half of 10 or three or 1.5 time 10. What's his overtime wage? $30. The two hours over time $15 per hour. You understand? understanding the terminology calculate the total amount Ryan for the particular week. So for the week, how much will he get? It's the 400 plus the $30 the 400. So this is just a basic question, simple numbers to show you what is going on. Of course, you're going to have to work it backwards and find how much hours he worked overtime without them giving you this. They can like tell you his total wage and you need to subtract the basic wage, get the overtime wage and work backwards, which is how it usually comes. We've done a lot of questions like that already. So that's the working. Oh, so here's something. If this if we say the gross income is this, the tax allowance is this and the tax rate is that, how much would be his taxable income? $80,000. Very good. So the taxable income is how much can be taxed. So you subtract the allowance. The allowance doesn't mean extra money. You don't add this. You subtract it. This means it passes through untaxed. So out of the 100,000, 20,000 is not going to be taxed. So it means 80,000 will be taxed. That's the taxable income. So what is his actual tax then? his actual taxes. It turn out to be 20,000 not from this as the allowance. It just turn out to be 20,000 when you find 25% of the taxable income taxable income amount of tax net income and the net income will now be 100,000 minus the 20,000. I do like how this example have the 20,000 show up twice. So maybe you should have used another percentage here like 20%. or no then it is so but 10% maybe so we should but the tax rate is usually 25%. So it's just a bad example in that you see 20,000 show up and I don't want you to think the 20,000 came from the initial 20,000 anyhow your gross minus your taxes will always give you a net income. So 100,000 minus the 20,000 taxes that we just worked out that'll be your net income. So that's how this salary and wages show up. Moving right along into section three just now. Let me see how my breakdown looking. Let me see how we looking for time there. If I need to speed up or slow down. What do you think? I need to speed up or slow down. Guess more speed. 2.25 hours. Um, so I have 20 minutes, 25 minutes to cover sets. I have 25 minutes to cover sets. Keep it steady. More speed. Well, some people saying slow, some people saying speed. So, I guess cuz we have the bad boy topics coming up now, right? These are fundamental topics. Um, we still have a easy thing with sets here. So, we kind of on par. We not too fast or we not too slow. Yes, we doing sets cuz sets come in paper one. So, I might as well mention it here. I might as well mention it here. Um, although the exam on Friday is not going to show sets. Explain concepts related to sets. I talked about this representing the set of. So, this is the set of even numbers. You understand what this means? If you have to list out all these numbers, what will you list out? Because sometimes this notation could show up in um paper 2D notation. This means all the numbers between one and five especially if they say x is natural. So it will be one 2 3 4 5. But do we get to include one? No. It must be greater than one. You could include five. So this is what they want. Understood? Set of numbers that are greater than one but less than or equal to five. So that's 2 3 4 5 and I need to add in this x is natural numbers otherwise this could be anything 2.3 and kind of consider this. This is the set of numbers 1 2 3. If I say if I use this symbol I would say two belongs to the set of a. We mentioned that already. Cardinality is the number of elements in a set. Okay, that they call it cardality. Okay, so the number of elements in A is three. That's this. That's this. U finite or infinite sets. A finite set is a set that has a specific number of elements. We notice like the set of number like what I just did the set of numbers between um 1 to five not including one. So 2 3 4 5 that's finite. But if I said the set of even numbers that goes on and on and so on that's infinite an infinite set uncountable number for example the the set of real numbers is an infinite set. So uncountable is kind of like the wrong word to use here because there are two types of infinities. There's uncountable infinities and there's countable infinities. What they call countable infinities meaning like you could you could map it out. You could keep going 1 2 3 4 5 6 keep on counting. You know what to do. Uncountable infinities would be like the set of real numbers where you could get 1.23 and you could just keep going endlessly with one set of different numbers. But nobody cares about that. So yes, this document is on the Google Drive. Sets doesn't come in paper too. Correct? But some of the set notation can show up. Universal set set that contains all possible elements. This is like what is outside of the ven diagram. And also some people will be coming back to this for the multiple choice. An empty set no elements. So the two symbols for empty set is this. And you all could write the next symbol. What's the next symbol? You all write it in the chat. Actually, you could write it on the computer. Complement of a set. The complement means outside. Jacqueline, I don't understand the question. Yes, this is the next symbol for um empty set. N is the symbol for number of elements in a set. So, if you have this and you have nothing in there, that means the set is empty. Jacqueline, I don't understand the question at all. You have to kind of let me know what you're trying to say there. Complement set is the set of all elements that are outside the set. So if U is this and A is this, then A complement would be 1 2 3 4 5 4a 5. That's all the elements outside of A but still in the set subsets. So we mentioned this. This is A. This is B. B is actually a this one is actually wrong. It's written back to front. B is actually a subset of A. I don't know why this written back to front. So B is a subset of A. I don't know if you're here for here. So B is a subset of A because B exists entirely inside of A. So you all take note of that in case Mr. Lane didn't get to change this in time. B is a subset of A, right? Not a is a subset of b. That's a typo. So represent a set in various forms. Set comprising the first three natural numbers. This is the set build notation 0 to four not including zero and four. So that's one two three. So the set notation is what shows up sometimes in the paper too. Like they might say show the region on a number line. And you all know I'm talking about we do this 100,000 times. What's that number line? 1 2 3 4 five. So this means I'm indicating numbers between zero and five. Like x exists x exists between zero and five. Sometime x is a member of the natural numbers or something like that. So we we can show this thing. So I don't want you to not understand what they're saying here. This is x exists between 0 and four not including 0 and four because it's not allow if it was equal then it would be equal. If it's equal you shade and this n stands for natural numbers. So natural numbers whole numbers we talked about at the very start. Listing this is how you list it. Subsets. What's the shortcut for finding subsets? If when you're doing the paper one, what would you use? 2 to the^ of n. Very good. So 2 to the^ of n where n is the number of elements in the set. 1 2 3. So 2 to the^ of 3 is 8. 1 2 3 4 5 6 7 8. Don't forget the empty set and the whole set are also subsets of the the whole set. You should be able to determine the number of subsets of a set with n elements. 2 to the^ of n. That's what I just mentioned. So if the subset was P Q R S, how many subsets do we find in this set? Just give me the number. How many subsets would you find here class? Stay locked in. Every time I sit down like other energies sit down to 2 to the^ of four which is 16. 16. So intersection. So um I'll just briefly mention this. We spoke about this recently in the last crash course. This is called the set equation. This symbol mean union. This symbol mean intersection. So what they are basically saying is let's do a ven diagram to see it. Everybody stay locked in. A B this shows up in paper one. The intersection represents this region. The union would represent everything in A and B. Right? So they're saying like let's say this is three, four and eight. They are saying the number of elements in A union B which is how much? And make sure you're locked in. How much? How much is A union B? 3 and 4 is seven. 7 and 8 is 15. is equal to the number of elements in E. How much elements in E? No. Seven. Seven in A. Three and four. How much in [Music] B? 12 minus the number of elements in A intersect B. Four. So in other words, if I say 7 + 12, I would have count the four twice. So I need to subtract it to find how much in all of them all together. So that's the application of this equation. You'll use it once in the paper one for sure. How much elements in [Music] A? This is A. This is A. So this is saying there are three. Yes. Cuz this is like if I have three things here and four things here. So 3 + 4 is seven elements in A. How much elements in B? So it's not like the number three is the only thing here in it. So uh this this type of set um this type of vend diagram is showing me the number of elements. This is like the number of elements in A. But I can do this. I can see like maybe there's a four, a six, a two here. Um I doubling up here. Let me just do this. But this is giving the number of elements in A. Number of elements in B. Yes, you add them up. So if you have the number of elements in A was three. In this A only that is like the number of elements in A only who study maths only or English only that'll be three. But the intersection here is actually four. So the entire thing was 3 + 4 7 five times just squeezing the multiple choice. Okay, so I just seen some people coming in last minute. Describe relationships among sets using set notation and symbols. Draw vin diagram. So all of these objectives here are tied into sets and I'll have to pull off a multiplechoice section um for sets some other day or just do some multiple choice on YouTube to help sharpen up on that before the multiple choice. When is the multiple choice exam everybody? Really? All right. So, we have real time sometime in June then. Lovely. So, I'm showing you shaded here some of the ideas. A union B, A intersect B. This is a nice way to kind of understand what's going on. Everything outside of A. This is everything inside of A, but it must at the same time be outside of B. Everything outside of A, but at the same time it must be inside of B. Get it? A and everything outside of B. Everything outside of A and outside of B. So in other words, this is also the same as A intersect B. Everything outside of that. So we want to shade everything that's outside of A and outside of B at the same time. A A A, this is a very tricky one. Yeah, watch this one, but it never really came. A complement intersect B complement. So where they both intersect would be this is like a union B complement. This comes regularly A union B complement. This idea today's teachers day. I didn't even know that. Thanks Haley. So here's a little past paper question that we snuck in into the 12-hour crash course that just kind of recaps the the last set of things that we learned. So basically we have learned number one. We have reviewed number one. Question one is done ladies and gentlemen. um calculate. I want you all to take a little rating of how well you can do this before we do these solutions. Can everybody here do this? Right out of 10 the exact value sharp people I want to see all your numbers. So you know the first thing to do is put it in your calculator then do it work it out right. Can you do your little conversion in EC dollars class? you know to convert. Then there's a compound interest question. Can you do the compound interest? So the the the consumer admitted can be one of the many different topics that we just saw. Wages, anything, anything. Bills, utilities, utilities in China these days. So credit union pays 8% perom. A customer deposits this in their account. Cal the total amount of money after 2 years. So you could just throw in the formula. Do see we did this already in a shantel. No, no audit are going to go through the questions. Don't worry. I'm going to and I'm going to show all the answers. So P1 + R over 100 N. This is the same formula that we we did earlier. So P in this case will be what? Chantel and my discord boy. You know that discord noise by Karen boy like ponging game boy. So P is 24,000. R is and I need to keep it on mute for important information. Certain things mute. R is eight. So you're just throwing this. What is N? Two. Cool. So you just throw that and you're going to grad. So let's see the solutions using a calculator. Plug this in. You get your 52. Double check that you can do this. carrying exchange this for that calculate the value of one basian to EC well if 2,000 gives you 2700 one fine for one will be 2700 divided by 2,000 and you'll get one Barbados dollars is 1.35 EC dollars everybody catch that cool so then she exchanged EC for Beijian dollars the amount of Barbados that she would receive. All I say this say Beijian dollar say Barbados I find Barbados that sound like a mouthful. So excuse me. So one represents $135 EC. I switch it around because I actually want to find out how much Barbados she's going to get. So I form for one by dividing. It's a division process. 1 one divided by 1.35 and I multiply by 432. I got my answer. So I saw that I was just I just needed to divide by 1.35 and I got $320. Quick maths. And the next part we substituted the formula as you all said. put it in your calculator and ensure that you get $27,99360. Okay, so that's question one. How you feeling about question one? Question one goes for nine marks. Everybody feel they could collect all nine marks in question one. Don't study that 11 marks. That was way back in the past. That's when I was doing math. Some multiple choice questions here. I'll skip all this for now to not waste time. Um, but it's in the handout. Y'all can go check it out. All the different sets questions that could come. Lots of them. It's all the possible sets question here, folks. So, here's what when I am. It's also on YouTube. Coins Springer sets multiple choice. I did this on YouTube. The reason I'm skipping it out is because multiple choice is not tomorrow. Why am I just coughing all cash to teach this can't this can't happen? Here's another pass paper. Elizabeth, you get you. Excuse me. Sharp people, please use brackets. Well done, Fate. Faith, you get the brackets. Very good. Here's another one. See what you all get here. See, I need to stand to keep my energy. Maybe that will make us 37 over 30. Are they get that? And they want the exact value. So, we're not going to do any funny business. Calculate the value of 1 minus cos of 37 correct to three decimal places. Put this in your calculator. See how if you get that out too. Don't forget to have a square here and then you get the square root of the whole thing. 0.602 0.602. Good. Standard form. How would you write this in standard form? I'm showing you all the answers just now. 5.27 * 10 ^ what -3 -3. So in this you literally need to write the square root symbol. For the sh they need to write square root then brackets 1 minus brackets again minus brackets again cos of 37 brackets square this close brackets for this whole one here. So it's like this closing for this you understand and this is what you'll need to have on your shaft to get the answer correct. So now you just need know that you want to find the square root of all of this people you just need to write it exactly like how you see it. square root. Let me show you this. This calculator I have here works just like square root one minus. So it will it'll keep going which you write cost 37 close brackets. Now we want the whole cost 37 to be squared. So let's put that in brackets. Put a squared there. Let's just get the open brackets first. only get 0.602 to three sig three decimal places. Thanks Josiah. So Josiah sends multiple choice questions there from YouTube. Yall can just take a little peek at that. Harish works at a call center 35 hours per week. He's paid an hourly rate of $11.20. Calculate the amount of money Harish earns in a four-week month. You want to do this? Yeah. In a certain week eight hours over time. Overtime hours are paid one and a half times usual wage. So you multiply this, you multiply, get the overtime wage and you find the total. So a very basic idea here. Very basic idea. Let's go take a look at the answers. Exact value. This is the working for it where we addition came first. Multiplication came first as well. So we did those two separately. Then we followed up with the subtraction and you got 37 over 30. For this one you all saw me do this on the calculator already. 0.602 5.27 by 10 to the3. Let's go check out Harish 1 week 35 by 1120 392 392x4 you get his amount for the four weeks in a certain week he works overtime 8 hours so 1.5 times the basic rate so this is the overtime rate and if I want to find the overtime wage I'll multiply the overtime rate the 1680 by the 8 hours over time which gives me this value one very good layout but that's not the final answer the answer was total amount paid but remember each week he gets this amount basic and then he did this amount overtime so you add the two together and you'll get the total this is past papers right not so Not so bad at all. Convert units of length. All right. So, we reach into what? Yes. Now I have to I have to follow the orders here. I not getting to each measurement. W not my turn. But hold on, hold on. I still have one other thing to touch for hour and a half. So in the next hour and a half, so we finishing my session like about 12:30. So from now till about 12:30, I am going to do vectors and matrices. Hang on to your hats. Let's go find vectors and matrices. I like vectors and matrices. Hey, real scrolling. Why? This is the only way to find to Isn't there a better way to scroll in this? Can I type the page number vectors and matrices at the very end? No. No. Can I not type the page number or something? Maybe I zoom out a little more. I just give a chance to scroll. CTF. This is not PDF for now. Is a drawing thing. I don't even have CT on this. I have a Mac. I type something here and turn off the whole thing. Click the link. No, the link don't work on this. This is that drawing tablet. Is it's not a PDF. this study. I scrolling through this fast and the other one revising. Um, ting ting ting notes. Okay, so we starting vectors and matrices and we will finish this maybe like around 12:30. So, Miss Nolini will pick back up with you all at 1. So, if I finish like 12, I'm really supposed to finish like 12:40 thereabouts. And you're supposed to get like a 20 minute break, a 15 minute break. So, 12:45. So, if I finish a little before 12:45, you get a little longer break. And Miss Molina will pick back up at 1 and take you all through for algebra, relations, functions, and graphs. um algebra relations functions and graph and measurement and I'm just coming back in for the last three hours basically I'll pick back up with statistics geometry and trigonometry all right people take a deep breath take a deep breath and let's go into vectors so vector is a quantity that has magnitude as well as direction however this is very abstract to What we're going to do with vectors, we're going to do very concrete ideas with vectors. So, explain concepts associated with vectors. Consider this coordinate. If you have the coordinate 43, 1 2 3 4 1 2 3. Yes. So, 43 somewhere here. Uh the coordinate represents represented as a vector. Like if this was the point P and this is the origin, this vector is O P also called our what type of vector would be 43. And this is a position vector. Well done. It's telling us to go across four. One, two, three, four. Then go up three. One, two, three. And you would have created the vector. Very nice. So the vector can be represented like this. Uh I have it as PQ. I started at P and then I went to Q. So it's 43. We can use a letter to represent the vector. You can use A or P, a common letter. Consider the point Ax in threedimensional space. O A X I + Y G O A A X Y O. So all of this looks like Huh. And I'm sure some of you when you're reading this is just turn the page and if you see anything like this in your Okay, I don't need to look. So um all they're trying to say is there's other ways to represent a vector. So like you see how I had 43, I could also represent it as 4 I + 3 J. The I here signifies the direction in X. I is a unit vector in the X direction. And J is like a unit. Unit mean one. This is like three Y's. Three one in the direction of Y. Right? So J is a unit vector in the direction of Y. So this is another notation. Admat students use this notation a lot. And in Cxc you just need to kind of know that is other notation. You don't really need to we don't really operate that notation. So consider the position vectors O A and O below. There's this law and what do we call this law class? A B is equal to O B minus O A. Yes, we call it the triangle law. So the triangle law says that if I want to go from A to B, I could go from instead A to O. So a to b is the same as a to o plus then go up o to b o to b. So anytime you're using plus the number the letters here must be the same if you if there's a plus if you're adding on the vectors to find another way. Keep the vectors find another way. So a to b is a to o plus o to b. These two things are saying the same thing. The idea is a to o is this way. O to a is this way. Understand? O to a So a to O is really subtract O to A the opposite direction. So that's where the negative sign comes in. And then we just rearrange this. So O B minus O A. So that whole line there's the triangle law there and it shows up almost every time in vectors. Um, so the powerful thing with the triangle law, those of you who have been doing your past papers in vectors, you start to realize why it's so useful to just know the triangle law instead of having to recalculate it all the time is because it ties a displacement vector to two position vectors. And we know that the easiest question in CCM maths is if they gave us B is the coordinates -3 5 and then asked you what is OB then all you need to do is do this. Why is any chance if you knew that? And if you didn't everybody know that right? So the position vector SA is taking the coordinates and just turning it vertical. So because these things really represent the coordinates secretly. If they gave us two coordinates and asked us to find the position vector, the triangle law is really handy. It takes that position vector and changes it to a displacement vector in one go. Fantastic stuff. Vectors are parallel if they related by a scalar factor. You all know that that shows up at the ending of the vectors question. So these vectors are parallel. This vector is saying go up, go across one, then go up two, one, two. So this vector is like this. This vector is saying go across go across two one two then go up four 1 2 3 4 right. So this vector is like this. Now because the ratio of across to upwards movement is the same. The vectors are actually parallel to each other. That's the whole reason why vectors with the same scale factor are parallel. is like it's the same vector but you just make it larger. the scale factor. Everybody, it's starting to make some sense now. Maybe I just memorized this, but now you see the reason why. So if vectors are related by a scale factor, they are parallel. And that's the proof that we use. There's another word that follows right after this. What is the word? I'll just kind of show you for a split second colinear. So M what I'm saying here is if I told you to go across one then go up two would you have created this vector here across one up two what about if I told you to go across three one two three so this is one one two and maintain the same ratio so one two three so now you have to go up 1 2 3 4 5 6 7 8 9 or no just six 1 2 3 4 5 6 so you start here end up there I would have created a parallel vector so this vector is 3 six this vector is one two but because it's just like multiplying this by three scale factor three they're parallel to each other making sense colinear happens when vectors are parallel but share the same point. So like if I told you this is A and this is B and I told you this is B and this is C. Well, hold up. If this is B, it not supposed to be there. If this was B, he had to be here cuz B had to be the same thing, right? And then C would be up there. So that means colinear. Colinear means colinear happens when they lie at the same line. So parallel means never meet. Parallel lines never meet. Same angle or disposition to one another. Spatial arrangement to one another. Colinear means lie on the same line. So the points A B C in this case are colinear because they lie on the same line. So they are parallel. Yes. Cuz the same same direction. If these lines were separate from each other, they won't meet. But they also lie on the same line. All right. Equal vectors. Two vectors are equal if they have the same magnitude and direction and is an idea that is used more often in matrices. But let's show you here like if I said 2 three and I said this vector is equal what is the value of what is the value of a? A has to be what? zero cuz this is two and a + 2 is zero. A have to be zero and B have to be B has to be two. Right? So in this scenario, what's the value of A in this scenario here? If these vectors are equal three, 2 by 3 is six. So we're just going through the ideas first before we start to combine everything into topics and vectors into questions. Two vectors are said to be inverse or opposite if they have the same magnitude but opposite directions. So on and we see this happening at vectors that go in the same um like if I reverse it a is the same as negative ba. Why is in the chat if you all understand that that idea is probably going to show up in the exam that kind of reversing through a negative sign. So if AB was 1 2 BA would be -1 -2 or if it was mixed like for example if I had something like PQ and PQ says to go across three but go down four that's PQ what wouldp be QP would be go backwards so negative switch here but then go up four so the sign switch it's not like it always turns negative it's It switches the sign. Well done, Zach. So simplify expression in vectors. To add vectors is very simple. You just add the the corresponding terms. So if I had that 2 + 5, I'll get -3 9. Everybody understands this. You add the corresponding term. U1 + U2. V1 plus V2. That's what they're showing you there. Subtraction. You subtract the corresponding term. So if instead I had a subtraction here, this was 2 - -5 is actually 7. 3 - 6 is -3. So you have to pay attention to negative numbers. In fact, what makes vectors hard is negative numbers and fractions, nothing else. Negative numbers and fractions is why people struggle with vectors. Do you know that the geometry part of vectors? Because if it was just simple numbers like two I am telling you if it was simple numbers like two three four one all the time everybody would have see what going on you see what going on but when it start to become 2/3 of this is 3 fifths of that and you have to add 2/3 x no 2/3 u plus uh one quarter v and that's where people is just start to spaz out so you just need to take your time and take a deep breath try to visualize what's happening so check the YouTube chat How to do it on the calculator. Oh, that's a long time ago. How to do it on the calculator. Vectors. Um, I don't think I ever teach how to do vectors in the calculator. So, I don't know what they're talking about in a YouTube chat. Now I show you vectors on the calculator. There's some stuff you can do with vectors and calculator matrices, but they're not that useful. Excuse me. So when a vector is multiplied by a scala blah blah blah, each component is multiply scala. So this lambda here physics people know that symbol lambda that's just a symbol we use to represent scala. Sometime I could have very well use k. So um all this is saying like if I have 2x 3 by -5 then it's the same as 6 by -10 which happens in matrices happens in matrices all we have an example here 2x4 is 8 2 by3 is -6 example two if I have half of 510 then it's 52 and the half goes of half of 10 is five cool right so if you given a point p is a b and they want the position vector. Please remember to just write it like this. It's the easiest thing you can get for CXC. Consider the point 23. O A would be 23. I mentioned this already. So we determine the magnitude of a vector. So the magnitude is like the actual raw size of the vector, right? And to find it you use a kind of pyth you use pythagoras theorem but you don't need to memorize that using pythagoras theorem you can just take it like this like if I have xy the magnitude and it's shown like this that kind of modulus sign it was going to be the square root of x^2 + y^2 so it's going to be the square root of whatever that is like for example if I have four and five is going to be 4<unk> square the square<unk> of 16 + 25 whatever that is understand class we take that magnitude one step further everybody take a stretch we take that magnitude one step further how do we take the magnitude one step further how do we take the magnitude one step further well there's something called the unit vector so if we want to Hold on. Hold on. Hold on. Hold on. Just one second. I tell you I was just don't worry I was just you. So the square root of 16 the square root of 16 plus 25. All right good you did. Where are we? So I was saying that this could go one step further. Yeah. unit vector unit vectors are found by okay first you find the magnitude like let's say I want to I have a vector Q let me just show you exactly right so if I have a vector Q and I said this vector is 35 and I want the unit what it will ask you for is the unit vector in the direction of Q don't panic the first thing you do is find the magnitude of Q which will be square square of 3 2 + 5 2<unk> of 9 + 25<unk> of 34 units not an exact number but whatever it is you could just put root 34 is there any no there's no there's nothing we can do in this is 17 by two so there's no s in this so it's just root 34 you can leave it like that and then the unit vector in the direction Q is going to be you divide each one of these by the the roo<unk>34 by the magnitude. No, Jacqueline, you don't need to you don't need to go and get the um decimal. You can just leave the square root 34. Yes. So you have some matrices ideas here. So you found the magnitude and then you divide each one of them by the magnitude. So really the unit vector is the vector divided by its magnitude and that's it. Determine the direction of a vector. Use vectors to solve problems in geometry. Okay. Now this where you need to determine the exact direction of the vector doesn't show up that much in this. The idea shows up more like in Um it shows up in physics and it shows up in um sometimes in bearings like where you need to find the angle with the horizontal not bearings um angle of elevation angle of um angle of elevation angle of depression. So you can get that kind of tan idea showing up there. But if it ever decide to pop up in vectors, this is what they mean. If they want you to get the exact direction of the vector in terms of the angle, which is not something that is very common, they want you to find the angle that it makes with the horizontal. So the idea that we've used to solve this is like for example, if you were given a vector, let's just say a random vector a is 2. This vector looks like this. You go across two and you go up seven. class, are you with me so far? That's a So, you draw that triangle and then to find the angle, some of you all maybe have an idea what you're going to do already. How do we find this angle? Well, that's a right angle triangle. This is the adjacent. This is the opposite. Which one we're going to use is the hypotenuse. We are going to use tan of this angle is the opposite over the adjacent. So the tan of the angle is 7 /2. More on this soon. We're going to do some tricks just now. Well, at the ending of this 12 hour tan inverse of 7 /2 and that's how you'd be able to determine the angle. So that's rare. Now if that shows up, you have an idea where which direction you will be looking at. Make sure your calculator is in radians. So in this case this vector is magnitude 5 and they gave you the angle is 30° anticlockwise from the horizontal as shown in the diagram. Let's just touch a little quick pass question here to kind of bring in everything. Now everybody take a deep breath and let's go. So the diagram below shows the position vector OP and OQ. OP OQ. Okay fine. What's the coordinates of P class? Let me just ask you some random things. Coordinates of P is what? Coordinates of P is 32. What's the coordinates of Q1 three? Great. So therefore, you can find your OP and OQ real fast, right? So OP is simply 32 and OQ is simply -13. Look, they literally ask you that for two marks here. Even in recent exams, they do that. That's what I'm showing you. Easiest question in Cxc. If anybody leave out that question, I don't know what to tell you. So that's 32. This is -13. The point R has the coordinates 8 9. So let's go find R. Okay, we can't find R. 8 N. Express QR as a vector in the form XY. So we want QR. How you think we're going to do that? Who we going to use immediately? What comes to mind immediately to find the triangle law? So QR the triangle law will be like O R minus OQ second one minus the first one right? Do we know OQ? Yes we do. Look it right here. Do we know O R? Yes we do. R is this. So O R is 89. Now you'll understand why I advise just learning the triangle real fast because it just ties in the two real quickly and then you just quickly 8 9 -13. So you end up with 8 + 1 is 9. 9 - 3 is 6. Class, how we feeling about that? Hopefully good. Using a vector method, show that OP is parallel to QR. So we just found that QR is 96. This is QR. Watch OP. Watch QR. Come on. How do we know they parallel? How do we know they are parallel? I just mentioned this and I surely think scale factor. Avoid the word ratio. Look, use the word scale factor. Can even use scala multiple. You see what is the scale factor here? No, use the word scale. S3. So you literally write QR is 3 * OP. So since they related by a scalar factor of three, they are parallel. Determine the magnitude of the vector P R. How do we do that? P R magnitude means find the square root of whatever PR was. Where P? Oh, we had to work out P R. We didn't work out PR yet. Why them so far? But I guess you get the idea now. So P R with the same as O R minus OP. We already know O R it's here. We already know OP it's there. You work out what it is and then we use the magnitude formula. I'll show you just now. I am telling you this style of vectors is very easy. You just follow the rules and you go through the type of vectors that seems hard or is really harder is the one with the geometry one with the U and the V and thing. And the only reason that is hard is because of the fractions and the negative numbers. No ads vectors is a different story all together. Sorry people, we're not talking about that vector is different. The point S has coordinates AB. Advance vector is not that bad either. Has the coordinates AB. Write QS as a column vector in terms AB. Lord have mercy. So S is AB. So O S is AB. And now they want you to write QS. So Qs real triangle law here. O S minus O Q which which we have and we have O S. So you substitute and go. I'm going to show you the answers just now. Given that QS is equal to OP, you equate the same two things we talking about. You'll get two things equal and I'm let's just jump into that and using a vector method show that parallelogram. Now this is a little more tricky. They want you to show that it's a parallelogram. So you're going to have to show that this line is equal and parallel to this side and this side is equal and parallel to that side. So let's see how we do this entire question. A very nice question to do over for yourself even today or tonight 900 p.m. This is one that I would advise to do over to just strengthen up a lot of different areas and vectors especially the triangle law. So here's the initial part. We did our OP because this is the coordinates focus people lock in lock in. We're looking at the solutions now. This was the coordinates of Q. So therefore OQ is this. You just switch it vertical. Express QR. So we did the triangle O R minus OQ. There we go. 96. We did all of that using a vector method. Show that OP is parallel to QR. We did this. We say that it's three times OP related by a scala factor of three. Therefore, they are parallel max. P R P R is O R minus OP. Substitute. Subtract 8 - 3 is 5 9 - 2 7 get our answer. Something jumping on my task. Let me see if I can get my off the screen. That distracting you. Come on. Come up here. All right. Good. I was jumping the guitar. I was jumping. So you're watching all my private tax man thing. CD that is logic as a music app. So P R is equal to the square root of five. Right? So what are we doing here? What are we doing here in this scenario people? We are finding the magnitude. So we got PR and we found the magnitude. Bang. This is the point S coordinates AB QS. Well, O S is AB in this direction. Using the triangle OS minus OQ, you get A minus1 cuz OQ is something that we had and OS we have it in terms of AB. So don't get frightened if you see letters you know that is maths there. So A minus1 is plus B minus 3. And now they said that this is actually equal to this. QS and OP are equal. So find the values of A and B. What is A? I'll leave A for you all to do. What is A? A will have to be two. And what is B? I could find B2. B will be five. 5 - 3 is two. Very nice. So there we work it out. And now to show that is a parallelogram. OP is 32. O P QS. OP is 32, right? I just let you all visualize it. QS is also 32. And we found PS is the same as OQ which is which is oh it's also 32 so that we hit parallel again. This is also 32 as well. O P and QS is 32 PS and OQ. Let me just make sure I not I don't I'm not convinced. PS was what? What was PS? We don't have PS anywhere. What's the next one? Just know people. Sorry for making it dy. I know the scrolling probably making some of you all dy O to Q. O to Q was O to Q. OQ was actually 13. So um this need to switch um NI or Josiah whoever three. This was -13. Both of these are the same. And both of these are the same. So what it's saying is that this is 32. This is 32. And this is -13, which is not how this would look. It would actually look backwards. -13 would be going up like this. So some kind of weird, it's a weird parallelogram. But what we're saying is that we have two sides that are equal to each other. two sides, two pair of sides that are equal to each other. These equal and these equal. So therefore, it would create a parallelogram. So that's your proof for parallelogram with vectors. Okay, moving forward into another example of matrices um of vectors. Should I start matrices? I have to I have [Music] to So let me start matrices and I'll come back maybe to the vectors example. Right? Are you okay with that? Let me start matrices first and then I'll come back to some more vectors example. So let's get a speed course in matrices. I I do a really nice neat video for the beginning of matrices for Tik Tok. I ran out of time because I actually just came back from Miami. I had to go and do some business there. So I ran out of time to do the intermediate. I did the intermediate, but I made a mistake in one of the questions in the negative. So I had to take it down. So those of youall who was on Tik Tok and watching out for that it may come up tomorrow. So but y'all you all getting the full thing those of you all who have in the cash and behold let's go through it. Yeah. Um where's the song from? Um this woman don't call me local international and we global explain basic concepts associated with majorities. 3 4 2 X the elements are members of the Okay, so these are called the elements. These are called the members. We're going real fast. This is a row. These are the columns. So this are 2x two row by column. So far so good. Why is in the chat all of these things? Long time. So the elements of the first row are three and four. See three and four here. Size represents row by columns. Two by four. Two rows, one column, 2x1 matrix. This is a 2x2 also called a square matrix because the number of rows and columns are the same. The identity matrix looks like this. More on this later on. 1 0 0 1. Just like before, remember the additive identity you don't change. 3 + 0 is same as three. Additive identity is zero. So if you multiply a matrix by the identity, it doesn't change. You get the matrix. Fascinating. Addition and subtraction. You can't subtract matrices that are not of the same size. But once they're of the same size, you can go ahead. So uh like if I have 3 4 0 6 + 2 -1 72, you just be like 2 + 3 is 5. You get three here, you get seven there, you get eight there. Ra understand this straightforward stuff. Very similar to vectors. Nothing nothing fishy about that. Equivalent matrices here. 2 a is equal to -10. So 2 a is equal to -20. Sorry. So a is equal to -20id by 2. A is equal to -10. So what you're doing is comparing the corresponding um so b is just six. You comparing the corresponding elements and you're able to find who is who. That's not let make sure my audio is the right audio. This thing always switching my audio. All right. Yeah. Are you hear me? Chris coming through the nice mic. Cool. Multiplication matrices. There's two types. There's the scala and then there's the matrix. Scala is what we did before with the vectors. For example, if I have a two sitting outside of a matrix the two is going to wet down everything inside here. You understand? So the two go and multiply. But we normally see scale of multiplication with fractions. So like if I half by let's say 3 4 -8 7. So in the end you will see something like 3 over2 half of 8 is actually -4 27 /2. Class are you with me so far. IU if you understand this type I for I understand. That's scala multiplication. You're just taking something on the outside and multiplying everything on the inside by it. Matrix multiplication is when the party matrices really start. Everything else was easy. This is when the party really gets good. This is when things start to get good. And matrix multiplication we do a row by column method. So let's say we have this 3 6 4 -1 1 2 and we need to multiply. Firstly you should know that this is a 2x2 matrix by a 2x1 matrix. Two rows one column. Two rows two columns. The numbers in the middle must be the same for multiplication. and two occur for multiplication to occur. These two numbers must be the same. The number the numbers in the middle must be equal. Meaning the number of row the number of columns because this is row by column. The number of columns in the first matrix must be equal to the number of rows in the second matrix. Write that down. Important stuff for multiplication. Right? And then the result is always going to look like these two numbers. So it's going to look like a 2x1 matrix. Its mommy is a 2x2 matrix. It daddy is a 2x1 matrix. This matrix will not come out looking like daddy 2x one. Catch it. So now let's do it. 3x one 6x 2 class. Are you with me? So we went we slid down this row and the same time we slide down that column. So the three go with the one and the six go with the two. That's the trick in a way. Yes. Genetic variation XY XX in a way. Yes. Correct. Kind of have that vibe. But this is maths uh four by one for of course of course I used to teach all these subjects and everything for I used to teach bio um chem physics everything that leads up to engineering so 3x one 6x two so we have three but 12 4 in fact if you look hard on YouTube you'll see me doing a crash course in bio many years of horrible quality That computer is real trash. So 15 over minus 2. Um but it's still good material if you want to like some stuff. 3x 1 6 by 2 4 bygative just now. Positive two. I talking too much. So this is two here. So thanks Michelle. Hey Zach. So everybody understand this? Is there anybody who not clear about this? If I have to do a 2x2 by a 2x two is is the same thing. See has these two styles. You have this style or this style. It's just that it will come out looking like well mommy and daddy's the same. So it come out looking like both of them. So you'll have a little more to do. So, let me just make sure you all know what's in get the answer here and tell me what is the value of A, B, C, and D. Literally write A is this, B is that, C is that, D is that. Let's do this real fast. Real fast. Let's go real quick. I don't really do much pausing and letting you all do stuff in these kind of things, but A is six. No, A is not six. Try again. Oh, sorry. A B. This one comes out looking like this. My bad. My bad. My bad. That's what I meant. AB. That's why he said A6. A is not six. Everybody tell what A is please. A is two. Good. And what is B? B is 20. Very nice. 3x 2 is 6. -1x 4 -4 2 10 by 2 is 20 0 by 4 is 0 2 0 K you see it 3 by 2 is 6 -1 by 4 is -4 6 + 9 Okay now we ready for this one tell me what's the value of a b c d let's go as fast as you Yeah, don't stick. We trying to go back and do some vectors questions just now. Yes, Karen. We had to do that too that in the next 5 minutes or so, Karen, we literally going to do everything that could come in CXD Max in this 12 hours. You'll touch on everything basically. I don't think there anything is going to get left out. A is -3. People getting A for -3. Anything else? B is 8. Let me see. B is not eight. How do you get B is eight? B is two. Very good. What do I get for C? B is not -2. B is just positive2. What do you get for C? C is -2. Very nice. and D is four. Very good. Very good. So those of you who get little errors, check out your check yourself before you wreck yourself in the people exam. Here's all the inner details. 4 by -3 plus 1 by -1. So -12 + -1 is -3. 4 by 0. Yeah. 1 by 2. 0 + 2. That's how we get positive2. Here you're going to get -2 here and 0 by 0 is 0. 2 by 2 is 4. Yeah. Yeah. Um just don't bother no calculator for this. Back in the days I used to show calculator for this and casio but it's take too long and it it almost harder to learn to do this and it is harder to learn to do this on the calculator than to do it on just do it. It's just multiplication. You can use it to check over. Fine. So m is equal to a b c determinant of m is a d minus bc. How many of youall know that formula? Nobody is in the exam going to tell you this formula. You have to memorize this. So um the adjint of a matrix is you swap the d and the e and you switch the side of the b and the c. So these are the two ideas that we use to find the inverse. We coming up toate Mr. Karen Jones. Wait, I'm missing something here. Objective nine. Objective 10. Define the multiplicative inverse of a non-s singular matrix. So a inverse multiply by a or James sorry Karen James. My bad. I had a Karen Jones in the class. So a inverse by a is I. Maybe like two years ago here. Karen Jones specifically a inverse time A is I the existence of multiplicative inverse of a non-s singular square matrix D. Somebody could read this and interpret what I trying to say here. Oh I know I'm trying to say here this is a question that's coming. So determinant of a everybody look back in the reason why I browse past this is because we talk about this already for anybody who saying like wow we just just mention and go we coming up to some working when we see in action if the determinant of a matrix class if the determinant of a matrix is equal to zero what does that mean? It's singular and singular means that you cannot find the radius find the determinant is zero. The determinant is zero. You can't find the inverse. Right? Good. So, so the determinant is zero. It's singular which means you can't find the inverse. Why have these big spaces? Maybe for me to do work I supposed to be doing work inside these spaces. Hm. Obtain the inverse of a non-s singular matrix blah blah blah. Let's do one. Let's do one. So, let's say I have this matrix 1 4 -6 5. Let's put this -2. The first thing I feel like I do this a million times. The first thing is to find the determinant of let's say the matrix is a is going to be a d minus b c. So -1 by 5 - 4 by -6. Why is in the chat if you're following this? We get 19. What a weird number. Oh, we didn't get 19. Sarah following Sarah as a question. Turn on your mic. Ask a question. What number? I don't know what number you don't know here. Is that number that you don't know how it came about? We lost. Um, do you know what this one came from? Sarah, do you know where the negative one came from? Where have you ever found the determinant of a matrix before? You haven't seen it in school, in lessons, in anywhere. Never in your life. You ever look it up? Yeah, you're blowing some people mind here S because that's early up. No teacher is there isn't a teach there may be some teachers are some s teachers me but everybody's teach this really so s I ain't trying to scare there may be hope so but you need to reise this s you're homeschooled you're in 405 and you never work out the determinant of a matrix and the exam is Friday you never ever ever work out the determinant of a matrix That's incredible, right? So, um, D A minus B minus C. This is the adjint. So, Sarah, the inverse of a matrix is, let me just teach it in Ariel exam tomorrow kind of way. These two numbers, you multiply them and subtract these two numbers. So, -1 by 5 and your minus 4 by -6 and put on your calculator and you're good. you're good to go. That's the determinant. It important to learn that. It's one of the early concepts in matrices. And if you learn the determinant, they could ask some little nitty-gritty questions on the determinant like what racial answer. So anyhow, I leave Sarah alone. So Sarah, the next thing is the adjint. So Sarah, what you need to do is look up matrices on YouTube. There's a nice little tutorial on that on YouTube. You can learn matrices. Matrices is the first big topic to collect marks. How many of you that was like the first big topic in maths? You learn that you can collect some marks on. That's the first big topic. After after that you'll start to look at computing the square and maybe bearings. But the first big topic in the section two to collect marks on is matrices. Not even vectors, matrices. So watch that one first. Sarah. Yeah. Sarah, you have you on team no sleep right now. you on team no sleep right now. So you swap these two, switch the sign on these two and then to find a inverse is one over the determinant by the adjint one over the determinant by the adjint. So it's going to be one over 19 and you multiply by the adjint. So anybody else doesn't follow what's happening here? We found the determinant. So this is the flow. You find your determinant. You find your joint and then you combine the two together and motion you could find the inverse. Yeah. So there's going to be some more examples of it. Just now we're going to do some pastels. Obtain the inverse of a non-s singular 2x2 matrix. You find the adjint. You find the determinant. And this is the formula. The inverse is one over the determinant by the adjint. Determinant 2x2 matrix associated with the specified transformation. This is where matrices get. This is Sarah. This is where everybody, not everybody, but a lot of people join you um on the transformation side because schools don't teach transformation. I try to do a survey here. How many of your schools didn't teach anything on transformation matrices like matrix transformation like how to get a matrix to switch something across the x axis and y axis? I'm sure some people a lot of people the school never teach that. Maybe a school taught you about transformation reflection but they they show you how to do it with matrices. Usually that's the case. So they just learn it. If given two matrices S and R and you asked to find the combined transformation. First big tip we're going to see how to do it in questions is now you all remember we did this in the last crash course in the April crash course and I was saying like if you have a composite function fg x which function comes first the g of x then you do it into the f. Same thing with matrices. If I need to if I have two transformations like transformation one and transformation two and I want to do this one then do that one on some point P then I have to say transformation one by transformation two back to front like this of P. So if I want to multiply the two of them to get one thing the second one comes first. Good. Um, time to do some past papers and start to see it in action. Now, Elizabeth just sent the matrices video. Excuse me. Calculate product 3 a b where a is equal to that and b is equal to that. The diagram will not draw to [Music] scale. Why isn't the chat difficult marks here if you was doing this exam in 2009? So the total math exam in marks is 100. I didn't see how much you had. I think you had 180 is 100. So they just want us to multiply these two things and multiply by three. I'm not going to waste time. I'll show you the solution for that soon. Diagram below not draw the scale shows a triangle ABC whose coordinates are stated. Bang bang bang bang or bang bang bang. Triangle ABC under goes two successive transformations. B followed by W. This is V. This is W. How many of you all can watch this and determine what type of transformation is about to happen? Do you know what type of transformation is about to happen? Not a reflection, an enlargement. Remember this is the identity matrix. If you wet down something with the identity matrix, nothing happens. The time you start to change up these numbers are true little negatives and numbers inside is when you start to do transformation. So if the numbers change from one to two you are enlarging scale factor two by the origin. Good. What is happening in this situation? Those of you all with the handbook a reflection in the y axis this is in the x point. So it's going to change the x value negative. So it's going to flip across here. So this is an enlargement. This is a reflection in the y ais. You see how useful those of you who are able to identify what it is. This can help you now to figure out what is going to be V followed by W. So V means get it bigger. Then go. So you know what kind of point what kind of triangle you're looking for. Looking for some triangles and all the X coordinates could be negative. It could be bigger. You all understand? Anybody cooking here or being cooked? Okay. So, state the effect of V on triangle ABC. So, it just said state. So, if you were able to um remember this, you could get away the two marks because it just said the state. Now, if you have to like show show the effect, then you'll have to multiply each point by the by the matrix like this one two. And this will eventually turn out to be two four. You can double check and see two 02. And you multiply by the next one one. This will turn out to be 22. Class you all understanding what I'm doing. 2 0 02 by 2 1 you'll get 4 2. So I'm taking each one of these points and transforming it using V. See the point here? I put it in like this vertical way. Multiply it and that's how you transform it. So all of a sudden the points are now 22. See this 22. That's B prime. Um 22 2. C is 42. That's this one. And A is 24. C prime and then determine the 2x2 matrix that represent the combination of V followed by W. So if I want to do V followed by W, how I'm multiplying W V good that critical critical and doing real transformations back in my days they used to do a lot of transformation had a whole period of no transformation now they reing down transformation on everybody. So WV - 1 0 0 1 2 0 02. Now this is kind of easy to multiply because a lot of zeros and a lot of ones. Just have to watch out for the sign. So I'm going to do it in one - 1 by two is -2 0 by 0. So nothing happens there again. 0 by two and then one by 0. So I have a zero there. And then - 1 by 0 and then 0 by 2. So I zero there. Then 0 by 0 and 1. This is also negative. Negative 1 by 2 is -2. So this is the this is the ending. And that makes perfect sense. Anybody seen why this makes perfect sense? Oh, sorry. No, no, this was negative. That was negative. It wasn't just positive. It makes perfect sense because go over this. What she said? I think she said go it over. We're going to show it over just now. I'm going to straight over the solutions. It's the scale factor means that it's going to enlarge but the x coordinate is going to switch negative. So this is how you combine the two of them. Okay. Anyhow, I'm going to show it over just now. Use your matrix in this to determine the coordinates of the image of the triangle. So what you're going to do is now take this just like what we did before and you multiply where you're taking this and multiply each one of the points and you'll get the final value. If it come like this that's like a straightforward matrix transformation. Nothing too it testing all the different things you could test but it's very straightforward. Write the following simultaneous equations. Ax is equal to b where ex this is your bread and butter. Anybody know how to butter their bread here? This is what my boy was asking for. Karen, James, Jones, whoever, Joseph, Karen, Joseph was asking for, right? So, this is the bread and butter. You could say everybody know the steps. We went through a whole class with this slow motion. XY 11695. First things you express it in matrix form. Then you write your little poem piece. Then you find your inverse using the terminal and a joint. Find your value of xy and you plug it in and then you'll be able to do this. So let's go and see how all these answers will look matrix product this by this. You all can verify this. So we got the product and then we need to multiply everything by three. 3 ab. So everybody going to get a little touch of the tree there. Max max then we move forward in the diagram below the effect of V we already established that it's an enlargement when you when you explaining an enlargement what are some of the important things to see we went through this last crash course the center and the scale factor center and the scale factor. So it's center is 0 0 because this is just a straightforward enlargement and the scale factor is two. So let's see if I got my marks here. ABC and V is an enlargement center and scale factor two max. So what program do you use? I don't know what I mean. This is a notepad. This is good notes or something. I don't know if it's that you're talking about. Determine the 2x2 matrix that represents the combination combined transformation of V followed by W. So V followed by W is WV. We multiply all of that we get our -202. This is the breakdown of the multiplication ratio. - 1 by 2 0 by 0 by two one by 0 and everything works out in the end to give you this. So what we're going to do is take this matrix that we just found was -2 0 0 2 and we're going to multiply all the points by that matrix. First point this is now your A prime. Second point multiply to get your B prime. Third point multiply get your C prime. So now we can even draw this if we ask to. It's going to look like something like this. Big up in the sky there compared to EAN down here. So far so good class and again a nice refreshers here, right? The following. So yes, so this is this is your bread and butter. This is one of the key big boy things you need to be able to do in majes. Um it takes a little while to study this. So I hope you would have seen it before and you're really sharp in it now. So you change it to the matrix form 11695 XY 67. You find the inverse of this. This is a little poem that I teach the students to write. So you can you can copy that program. Yes, that's totally correct. So you you copy this then you'll find the inverse of a. So a we already did how to find the inverse determinant first bang adjint one over the determinant right they joint so the determinant turn out to be one which is fantastic but let's say the determinant turn out to be seven do we put the seven under everybody do you put the seven under everybody in this scenario no if they ask you specifically to find inverse do it. Put the seven on everybody. If not, don't do that. It's just going to make everything messy. Keep initializing. Zack, why am I Yes. Or the whatever it is. Why my computer trying to restart it? And hold on. I don't even use Windows. So why them update? I don't know what going on there. So one over one in this case you leave everything on the outside but it turn out to be one so it's not affecting anything. No anytime things going strange on my computer is called that mean. So if this is 1 over 7 you leave it here. Do your multiplication first ratio. So 5 by 6 and then -7 by -6 by 742. You work it out and -9 by 6 11 by 7 is 77 and you add it up and then I end up bringing you a 1 over 70 and you put everything over seven but it turn out that is one. So it didn't have nothing to do. So this is it here. I don't feel sorry for Zach and I feel sorry for me. It wasn't here whole year. So x is equal to that typo -12 and y is equal to 23. Josiah Oralini if you're here type over there type. All right. So let's do another one. Do I have a next one in the chambers? So we're going to do our next one of this. Then we're going to do another vectors and maybe another vectors or maybe another matrices. It depends. But we stopping like around 12:45 to give a little 15 minute break and Mr. Lin going to come in and handle the next like five hours. Wait. So the matrices A and B represent the transformation given below. A is this, B is this. Let me take a little survey. How many of you all know right now? You know it. You already memorize them. The different transformation matrices like you know watching this this is a rotation 90°. This is a reflection in the straight line y=x. Kavir only. Cavir is the only man in the whole class. Elijah, I gave earlier this in the notes already. If you're in my class already, you could go and memorize it. Those of you who have the student formula booklet, I turn on No, I didn't do it on an admin sent me um something on Tik Tok and see my whole formula booklet in 3D. My whole formula booklet on Tik Tok making the wrongs. I just say boy. So if you don't have the formula, make your way to the CC math formulas or whatever you can see. So these are two types of transformations. Lolita Lolita not today not today. So these are two types of transformation two types of transformations that you could just memorize one time but they're telling you determine the elements of the matrix C which represents anticlockwise rotations of 90° about the origin followed by a reflection in the straight line y=x. Why is in the chat if I can do this? Yeah, I see myself on some of the be in class people is snap it and wait. I see a parent on Facebook say inappropriate inappropriate man. That same one gentle that same one. Determine the elements of the matrix C which represents an anticlockwise rotation of 90° about the origin followed by a reflection in the straight line y=x. What they trying to tell you here? What they trying to tell you here in part? Yes, J. Hey, this man real smart boy. I hope you bring home that merit list for bring home that merit list for J. Um, so which represents so you all understand. We see this already. Well, I see this. I just see this here. That A and boom shot. I just see this here. I B. So they're saying A B. So they want us to do A then B. But remember we can't we have to do back to front. We have to do B then A. So they're basically telling us to multiply B by A. You see it B by A. Those of you who joining the non-sleep club, make sure take some power naps in between. Otherwise your brain will get toast toast. So zero negative power naps in between. B A 01 1 0. So let's see what happen when you multiply B by A. So 0 by 01 by one that goes here. So that's negative 1. Wow, it come back up. 0 by 01 by one. Okay. 1 by 0 and 0 by All right. So zero goes here. No racial you know this. So this is positive 1 and 0 by 0. So this is what happens. This is the answer here. Describe geometrically the single transformation represented by C. So what is this? Some of you all could just watch it and tell me what it is. We actually experienced it already. What is that? We did it today. That is a reflection in the y- axis. Moving right on along. So another transformation. So I'm so glad I'm seeing a bunch of transformations here. So we're kind of prepping you for that. Is defined by the following 2x2 matrix where K is a constant. So this is the transformation. Is this a known transformation ladies and gents? No. That is some kind of random thing 2 one negative 1. What in the world is going on there? So we can't do anything about that. It's not like we have a have that memorized. So we have to go forward. T maps this to this. So what what does that piece of information tell us? It tells us T multiplied by 23 gives us 815. How many of you know where this question is ending up? Why is in the chat if you're already smelling the the result of this question here? So T is multiplying by this point to create the image. This is the object. This is the image. If this was point P, this is P prime. We're going to have to determine K. Yeah, we're going to have to replace this with 1 2 K1. Multiply out things and see if you can figure out K. What's going to be light work? Determine the value of K. So, solutions time. Solutions time. So, C is B A. We already multiply that. We get one 01. Wait, wait. What paper is this? June 2023. two years ago folks. So um matrix C represents a reflection in the y- axis. This is t k stuff. We multiply. We put down this. Everybody look at it. TP is p prime. This is the object image. Object image. Good. Everybody understands what the transformation matrix is doing. How it works? Really shout out to um Admin Cada for sending um the breakdown so you all could take a look at the breakdown so you can see where we are at. We are now finishing vectors and matrices that first section there. Algebra is coming up at the top of the hour from 1:00 we'll be doing algebra then relations functions and graph then measurement which will finish sometime around 6:00 or so and I would be coming in with statistics and geometry and trigonometry to wrap it up you're not paying attention so if I have a me if I have this point here, right? Boom. At the point and um a transformation come this point T. To get where the point will not end up, you do this T by the point and you get a new image. So like if this is a T by A is a prime or a complement or whatever you want to call that right now and watch this. If the point was attacked by two, like for example, I want the point to undergo the transformation T then followed it by followed by the transformation P. You don't write T P A. That's wrong. Who comes first in terms of writing the P comes first cuz the T gets to attack the A first. get what that is and then the P. So that is why B A and not AB. It's it's very similar. I said this I said this before as well. It's very similar to the idea of composite functions. F GX is what you're going to do with it miss. If I said g of x is x + 3 and f of x is 2x and I said find fg of let you put x let's put five what I need to do first this is this is telling me yes the g first why because the g is closest to the five so if I'm saying two transformations occur a then b and these things occur on point P then I means I have to write like this. A will occur in P then B. A then B. Making making sense. All right. We got to go forward. So we multiply. Now we don't need to multiply everything here. People look at the intelligent movements here. Look at the big brain movements here. This by this will give me this. Fine. I don't need to do that. And that's not part of what I need. This by this will give me 15. So I'm just going to do that green highlighted spot there. Two by K or K21 by 3 -3 must be equal to 15. And from that I can work out what K is. K is 9. Well done Lita. K is 9. Hey the last page of the notes there. So let me go back up to the vectors now and knock off a couple vectors questions. So that's matrices and that's what's coming in matrices. Now we we see we kind of covered everything in matrices. The determinant that joint inverse simultaneous equations but we did a bunch of transformations here. The transformation idea is what is coming now class. So you want to focus a lot on transformation. Make sure you can still do the simultaneous and stuff but be prepared for the transformation. Every hour somebody asking if the notes in the drive. Yes, the notes will be in the drive. It's not in the drive and they could go and protest on the WhatsApp groups and thing. We want the notes right now and they will send it to the drive. No, but they're supposed to send it just like the last crash course. They'll send it to the drive really quick. And those of youall who want to watch over the recording, usually the recording takes like a day, a day and a half because it's so big to load up. But but because I'm streaming it live on YouTube as well, the YouTube crew, which has 53 people on it, by the way, YouTube you all could just um the link will be there, right? Once you have your registered email, you can access it. Karen, it don't even show up by 10 and 8:00 p.m. So, the reason why it doesn't come up instantly is because after this meat is finished, it processes and then it pops up on our drive and then we need to download it and upload it and all kind of things. So, and real persons have to do that. So, that's why any one of these 12-hour crash courses kind of important that you stay in the crash course. you're coming live because the recording is only going to be up for a short. All right, let's move forward. So the following vectors are defined as shown blah blah blah blah blah blah blah. Determine each of the following a vector other than 5gative1 that is parallel to wx. So this was where this was a recent paper in June 2023. Yeah, this was they don't usually give open-ended questions. This is a open-ended question meaning I could literally write anything. That is correct. I could write 50 and -10. Feel like I do this recently. I did this in the other cash cost. 50 -10. It's open-ended. So Wy. So let's let's skim through this one really quick since I did it in the other cash cost. How would I find Wy? And how would I find X Hey, more triangle laws there, right? So, we just hitting them some triangle laws. Or we could use we could do the first principle thing where we say like W to our next one. Maybe W to X and then X to Y. The reason why we going to use this and not the triangle is because I don't really have a and I think that the reason why they bring it like this. This here I don't really have a O here. You seen any um O's there? There are no O's. You all see that we don't really have a um what am I trying to say? We don't have a position vector. So you have to use the key to vectors which is finding another route. So anytime you're using the addition, I said this earlier, these two letters will be the same the letters on the inside because you go like it's like you're going from W to X and then you pick back off an X and you go to Y wy wy. You all visualizing that properly? So like instead of going from W to Y, I go from W to X and then from X to Y. The key to vectors is finding another way, another route. So w y is the same as w to x and x to y. No colinear Jonathan colinear means line on the same straight line. Wx and y on the same path if you me but it's not the same straight line. It doesn't have to be colinear. So it only colinear if it lies at the same straight line. Okay. Okay, so this is xz and this is the what this means what they asking us to find here the magnitude the square<unk> of -3 7 well done then Liam so this this is literally a standalone question because it can give us that there even if you unable to find some of the things this is magnitude this is just a formula that's what I want you to approach all the questions in this paper with don't see vectors and like some of y'all here s who never do some things in your life before. Don't reach to the question and be like, "Okay, I don't know how to do this. I don't know how to do this." Skip out this whole question. No, check each part and you might see, oh, I oh yeah, I remember the formula for magnitude. That just means I need to find the square root of the -3 2 + the 7^ 2 and you collect a two marks in a very important question, right? Still get a little two marks there. Okay, so this is the answer. A vector parallel to this is 10 -2. I just made up one and since the scala factor is two. This is the solution how we find w y go from w to x x to y. To find xz we go from x to y and y to z. We didn't have y to z. We actually have z to y. So we did we did the negative. So -3us 8. So y'all can check through all the working. I'm skipping skipping all the little arithmetic working right for time sake so you can get the next question. Um the magnitude same thing and it's not just 58 is square root of 15 right Kai square root of 58 is 7.62 62 units. Three significant figures. Moving forward. Wait, wait, wait, wait, wait. Now, there's something that was left out here in vectors that don't I'm not too happy that it's left out. So, I have to do I have to have to get in one of that. What What was left out? Not happy about that at all. The geometry part of vectors. So, you all let me just squeeze in that and we'll go on the break, right? Just give me a little chance. 2025 Jan. Was there any geometry in CET maths 2025 Jan? Those of yall who did the Jan pass paper I am not seeing yes I oh we did this in the um last crash course kind of disgusting. I think we did this in the last crash course. those from the last cash. Correct me if I'm wrong. If we did it, I'll find another one. We do this one. Yes. Yeah, we did. Kadia, the YouTube link that I sent, you just click on that link. It's as simple as that. But make sure you signed in with the email that you're registered with. We didn't do this. Some people say we didn't do it. Okay. Okay. We didn't do it. All right. So, I did a reun it. Okay. So, the diagram below shows quadrilateral OM in which O is the origin. So, origin there and O L is 4 Y, O M is 6 Z, O N is 2X. The point A lies on LM such that LA A to AM is one to two. So how many parts is split this in this line? Three three nice parts, right? So I could put a little dash there, a dash there. So A to M, L to E, L to A, see it L to A to A to M is one to two. B is the same idea. M M to B is 2 to one. So I'll split this in three as well. Now there's a way other way this can come. For example like if I have a point P here and I said A to P is A to P is the same as A to B in this ratio. It is equal to 1 to 4. How much part to split this in? So all those fives are not correct. So this is the other way you can come. So watch out. Right. This one is kind of straightforward. They said L to A compared. Watch watch what is showing you here. L to A to M. So L to A one A to M two parts. Natalia seen why it's four. Now Jen why it's four. This one is saying A to P as compared with what? As compared with A to B. And now I need you all to know that this each one could like 50% chance they can they bring in this ratio. You can either bring it this way or the other way. And students get wrong all the time because of the ratio. So in this case, this is how much fats? Four parts. You all see why it's four parts. Now A to P is like one and A to B is like four. So it's saying A to B is four parts. See the subtle difference there. Instead of like having like A to P and P to B, if it was A to P and P to B, then yes, you split it in five parts. Subtle little trick there. Those of you who do the past look f you'll see it. Um anyhow, so what what we wanted to find MN in terms of X and Z. [Music] MN I feel like I do this recently. Maybe I did this live or something. I did this recently. MN MN What do I know? This B isn't like a U or anything. I need to find another way. So if I find another way, it will be 6 Z. But hold on. I'm going opposite 6 Z. So, MN is the same as opposite 6 Z and then this direction. Oh, let me write it out first. MN is the same as M to O, then O to N. Add it to Tik Tok. I tell you the other day, okay, - 6 Z + 2X. Then you could rearrange it. If I'm trying to find, let's see if I do one more piece of this. Ln L to N. I'm trying to find L to N. At the ending of the crash course, you can pick back up some more of this as well if you all want. L to N would be what? You're going down this one. So, so - 4 Y + 2X. So I could rearrange it and write 2x - 4 y. Right? Usually they don't put y and x. They put u and v. And usually it's not such a complicated shape. Usually it's just a parallelogram or a triangle. So this was one of the worst case scenarios. You all could take a look at this. This is Jack 2025. Mel, you lost. This is based off a very simple idea. Mel, what remember what I told y'all? The reason people get cooked on this topic, no, you can't do it another way, Zach. The reason why people get cooked on this, that's like saying one plus one, you can add it another way, you're going to get two, right? Um, the reason people get cooked on this topic is because of all the fractions and negative numbers and thing. It's a very simple topic. Like if I create a parallelogram and I say this is tree U and I say say this is maybe U and I say this is A B C actually I have to say this is D ABC D and I ask you what is the value of um AC what is the answer? No, Aiden. This is free marks. Aiden key is the key to vectors is finding another way. If you want to go from A to C, it's like you go from A to B and then B to C. So you could write that as like your starting. Cool. And then you're filling the blanks. A to B is V. Not really. It's opposite the R. I just switch it. So that's why everybody say negative. And then B to C is 3 U. And that's the end. Or you could rearrange it and put 3 U minus B if you want to have the negative kind of hidden like that. Usually that's the Yeah, usually that's a convention. We don't say U3. We say tree. That's a convention as well. We put the number before the letter. So our convention is if you have a negative and a positive we normally write the positive term first. Um now if I want to get like midway here there's a point M here midway and then I say I want to go um MB figure out what's the value of MB. This is where it gets a little confusing for some people but midpoint of this means that if this is 3 U this is the same as 3 U. So this part will be 3 u / 2. Key to vectors is finding another way. So you have to go from m to a and then from a to b. So it's going to be m to a is opposite. So it's -3 u and 2 or some people have it as -3 over 2 u plus a to b is minus b as well. So this is what you're looking at and you're cooking class. Bam. And then maybe if you have this one and uh this is uh S this way going this way S to D is the same thing but this time it'll be positive and then you can show that they parallel after two years. So you'll go review this vector stuff again. Um you could if you have a vector question recently that you real gears it up. Um most likely I'll finish with a little buffer time and we could do some of the toughest questions because when you finish at the end of today like maybe 8:30 I'll finish 8:45. I'll have a little 15 minutes to pull out some of the toughest questions you'll ever face. But in the meanwhile it is 12:42. The time is flying. We started 9:00 and look at where we are already quarter way through. So for this period from now till around 6 something 6 something I don't I can't remember the exact time it's finished last year. Uh Miss Nolini will be in charge of you all and there will be a little period here 15 minutes for you all to have a little lunch or whatever or maybe not I don't know or to just recap some stuff while we eating. You could go and check through some of the notes as we eating. Um I think Mr. I think you could send them the notes. Oh no no no can't send the notes yet. You have some solutions from yours or do they have the student version of the notes yet? Just make sure Miss Nolini can stream properly here. She's coming back from 1:00. Yeah. So, Mr. near Here it's one document with questions and solutions or by sending it already or we sending it after I sending it now makes sense because I have some questions that come up in it that I want to ask you but I mean I can always add limb the question I can I could I could add lil the question no but then each after each topic we We have the um we have the pass paper. So we'll send you all it after class. Sure. Zach, you could send it. We'll send you all the um notes after class. So the break starts now. It's a 15-minut break. I'm just making sure Miss Nolin can present for the um for the YouTube. Mel check out to see if the YouTube presentation is working. So YouTube Yeah, we on a break. Yeah, we on a break. We on a break. All right, at the stage I have to remove mine first. Right, there we go. Um, so you got three there, right? Um, just double check your mic for [Music] YouTube. Aon, so you just need to click the YouTube link to get it. Aon, you just need to click the YouTube link to get it recorded. Nalini, you hearing me? Oh, definitely hearing me. Oh, you hearing me? Oh, yeah. Double check your mic for um for YouTube. All right. Check in. One, two, three. All right. Yeah. All right. So, and what about this YouTube link? Um, all right. Good. So, I'm this Google Meet link. So, I'm turning off my pres presentation here. So, you could take over the presentation on Google Meet as well. I just want to make sure I transfer properly. Right. So, YouTube people should be hearing me now. Right. You can just drop a line in the chat if you're hearing me on YouTube. Right. So, all I said so far was that the difference between an expression and equation is the presence of an equal sign. You should be use what is a variable? You should be able to use variable symbols such as X Y Z. So any letter of the alphabet A, B, C, D to represent unknown quantities. So variables they can vary. They can change. You should also be able to use inequality symbols. So you have less than, greater than, less than or equal to as well as greater than or equal to. Right? So symbols, numbers, operations, etc. You need to be able to translate between symbols and worded expressions. Basically, what that means is you taking the English and turning it into maths. So, if I tell you four is subtracted from the product X and Y. So, everybody find your keyboard, type what do you think that is in terms of algebra. Four is subtracted from the product. When you hear the word product, you're going to take think multiply. So, try again. Um, I mean not not lucky. Jonathan, try again. Four is subtracted from XY. So you're going to say XY - 4, right? Four is being subtracted from the XY. So you first write XY then you write -4. This is another worded statement. Similarly along the lines of like linear programming, the ingredients of each small pizza cost $15 and for each large pizza $30. Right? Underline important information. The manager of a pizza shop wishes to make X small pizzas and Y large pizzas. He plans to spend no more than $450. So this is giving an inequality vibe. What should I use to represent this information? Right? So, X small pizzas, so that's $15 for a small pizza. So, that's 15X. $30 for a large pizza. We have Y large pizzas, so that's plus 30 Y. And I only have $450 to spend, so it will be less than or equal to $450.4 450 is my max less than or equal to 450. So it cannot be greater than that. It has to be less than or equal to that. So let's look at a past paper question which we will um which we will take the English and turn it into maths. So write an equation in x to represent each statement below. When four is added to a certain number, the result is the same as having the number and adding 10. Right? When four is added to a certain number, you can let the number be x. The result is the same as having the number and adding 10. So what should I write? Is the same as having the number means multiplying by half. Having the number half x very nice, Jonathan. Half x adding 10. All right. So the statement is 4 + x is = a half x + 10. So we don't need to solve because the question said don't solve. Let's try another one. Squaring a number and subtracting six gives the same result as doubling the number and adding nine. Squaring the number. So if I square the number, let's call the number x. If I square it, I get x squ and subtracting six gives the same result as doubling. Doubling the number means 2 * x and adding 9. So we have x^2 - 6 is equal to 2x + 9. Very nice everybody in the chat. So objective three you have to have to know your directed numbers. So there are two rules. One rule for multiplication division and when two signs are next to each other. So rather talk for me when you have a plus and a plus that makes a welcome. Welcome. So yes, a plus and a plus makes a plus. A minus and a minus also makes a plus. But when I interchange the signs, a plus and a minus makes a minus. A minus and a plus makes a minus. So you know this is a everything crash course. You're making sure to do everything, but everybody has to know how to do directed numbers. Please understand that this table is not for everything. There's an entirely different rule for when you are doing addition and subtraction. So the first part of the rule says check to see if signs are next to each other and if they are change to one sign. All right? So if I had -2 + -3 the two signs are next to each other. So I have to change it into one sign. If they have the same signs we add the numbers. So if I have -2 minus 3 I add the numbers and put back the sign. What's the answer? In our minds, we do 2 + 3 is 5. Put back the sign, it's -5. Very nice. But if I had different signs, like if I had -2 + 3, those are different signs. So you subtract the numbers without the sign, put back the sign of the larger number. So -2 + 3. In your mind, 3 - 2 is 1. the the sign of the larger number is positive. So we don't need to put a positive one. We simply put one. Lovely. Right. Moving along. We are going to learn how to simplify algebraic expressions. A must know in seat maths. So you can only add or subtract like terms. They must be like each other. So question is x and x² the same thing? Do you all think x and x2 is the same thing? Absolutely not. X is a straight line. X2 is a curve. Those two things are different from each other. And you cannot add them. So if I give you a question, 2x + 3x + 4 y + 5 y. All right, this is the algebra section. I want to see a lot of answers in the chat. We're going to add it. 3 and 2 make 5x. 4 + 5 makes 9 y. X and Y are totally different variables. These questions are often mixed with the distributive law. Right? So if you ever checked out the past papers, you'll see that. So taking a look at 2016, they they mix these questions with the distributive law. Right? Making sure everybody inside. So if it is I ask you to expand and simplify. So Mr. Swing went through the distributive law with you guys. 2x x * x is = 2x^ 2 x * 5 10 x. Make sure to take the negative with you. So you're going to ask yourself -3 * x is -3x -3 * -4. It's right pos2. And the question also said simplify. So to get all of your two marks, the x's colline with each other. So you get 10 - 3 is 7 x + 12. Very nice. So you want to you want to make sure and simplify 10 - 3 7 x. We must of course know how to substitute numbers inside of algebraic expressions. So my algebraic expression here is 2x y^2. You are given that x is 3 and y = -2. So you write it down and you make sure to show your substitution line. You're substituting x as 3. Please use brackets -2 to the^ of two. Please use your brackets. If if you don't use your brackets, they might be trying to do incorrect things. So use brackets. So 2 3 -2 all squared everybody is is pos4 right that's paper one things there. So 2x3 is 664 is 24. Very nice. So some tips whenever you're doing questions of substitution specifically even in measurement you get substitution things happening. Show the substitution line. Usually one mark is awarded for that. Write your solution going down the page. Put only one equal sign per line. I'm sure you've heard these things before. So just you know wrapping it up. It's a possible question. Given that a= 4 and b = 2 and c =1. Find the value of a minus b + c. So you're going to substitute a as 4, b as 2, right? Every time I substitute something, I put brackets. Plus C as1. So being good with your directive numbers. That's right. 4 - 2 - 1 is 1. What if I ask you to find 2 a to the power of b. So I just want to make sure everybody understands that this is to the power of b. So we get two a is 4 b is 2 4 to the power of 2 is 16 2 * 16 is 32. Next objective so this is what we call fancy substitution. The main topic is called binary operations. So I'm giving you a binary operation takes in two inputs and produces an output. So the operation says a* b is equal to 2 a minus b. I want to find the value of 2* 3. So when I compare it right everybody what am I substituting A as when I compare it I'm substituting A as two and I'm substituting B as three. So you want to make sure you compare it properly. Order is important out here. Right? So 2 * a as 2 - b as 3. Right? So so you work it out. What do you get? So take your time. 2 4 - 3 you get one lovely. So binary operations is really tested but it can come. It's in your syllabus. So if I give you the binary operation star right they give it to you and you need to find the value of three star for. So when you compare it what are you substituting a to b a will be three substituting b to be four. Very nice everyone. So when you put your make sure to put your substitution line. A is being substituted as three. B is 4. 2 * 3 * 4. So we get 7^ 2 - 3x 2 is 6. 64 is 24. So what's 49 - 24, right? The people in chat is killing it out here. That's right. It's 25. All right. So let's just make sure we know the distributive law. We use the distributive law to factoriize or expand algebraic expressions. So what it says is that you're taking everything outside the brackets and multiplying by everything inside the brackets. So a by b plus a by c. So let's just look at a multiple choice question. If I tell you 49 by 17 plus 49 by 3. Very nice Elijah. How do you making sure the wheels turning in your brain? Understanding algebra. Algebra is tested throughout the paper. It's specifically tested in question two, but it's a foundation that's tested throughout the paper. So, you have to know it. A B C D everybody I want everybody put in an answer in the chat. All right. So, if I factor out that 49, I get 17 + 3, right? Because 49 multiply by 17 + 49 by 3. So, remember you are Multiplication distributes over addition. That's what you're doing. So we get 49 ultip by 20. Answer choice A. Lovely. Or you could do the other way. P * Q minus P * R. So it's factorizing. What is common? P. What is missing? Q - R. So you get P * Q - P * R. So answer choice um D right everybody D I see a C hit the chat just just looking back. So you're factorizing. So the distributive law, right? So what I ask myself Elijah is what is common and what is missing. So when you look at the expression, do you realize that P is common to both of them? No, it wouldn't be C. So P is common to both of them. So that's what you're factoring out. And then you ask yourself what's missing. So P * Q minus P * R. So the answer everybody is option D. Take note of that. So we look we go into factorization in a minute as well. Algebraic fractions. So for algebraic fractions you do need to know what is the LCM. If I asked you the LCM of 6 and 8, what would you tell me? A little less than that Jonathan. A little less than that. So a lot of us would be like 6 * 8 is is 48. But there's a factor of two hidden there. So I really want to use the 24. So Jonathan, even if you use it, you're making sure to simplify in the end, but you want to really use the 24. Nice. So you ask yourself, all right, this is your flow of movements, everybody. Six can go into 24 four times. Multiply by your numerator. So Carol you could use the 48 but when it says that you are writing your final answer you have to simplify. So you do end up factoring out two and and and all of that once you simplify at the end you will get it correct but you really just want to avoid the work and use 24. You go again 8 can go into 24 three times. Multiply by your numerator x. Make sure to bring down that minus sign. We expand and simplify. 4x + 4 - 3x all over 24. You don't leave your answer like that. You simplify everybody. 4x - 3x is x. So we get x + 4 over 24. Question. Can I do any more simplification here or am I finished? I'm finished. I cannot do any simplification anymore. If it is your thinking things like that, no. Absolutely not. We are not doing that. I've seen that too often where students mix up mix up ideas behind simplifying. You can only do things like that when it's separated by multiplication. So, where did the 3x go? 4x - 3x. I simplified it to x. So everybody this is your final answer. You cannot simplify again. That is it. And there are questions where the LCMS where the denominators are x and x². So what are the what is the LCM of x and x²? So Shantel has it x squ right everybody the LCM of x and x squ is x squ. So if I had it's like a multiple choice question with this I can't I can't remember it. I think it's like this. So if I had 5x and 3x exactly. So you want to make sure you understand the LCM here is x squ. If I have 5x and 3x everybody, what is the LCM? There's a a popular multiple choice question with this. The LCM of 5x and 3x. So five and three the LCM is 15. X and X just simply X. So you ask yourself, how many times can 5x go into 15x? Three times 3 * your numerator. How many times can 3x go into 15x? 5 * 5 * your numerator. So you get 15 over 15x. That's a popular paper one question. All right. to next. Make sure it's sticking. Um, all right. So, let me just unpresent and present over cuz that is supposed to be happening, right? So, I'll just present again making sure everything is good. You should be able to see what I'm writing as soon as I write it. So the laws of indices, Mr. Spinger looked at the laws of indices already. You want to make sure you have a proper proper understanding of the use of the laws of indices. Right? So the first one says a power of multip is equal to a the^ of m + n. In English, what that means is that when you have the same base, you add the powers. 2 + 3 gives five. When you are dividing indices of the same base, what are you doing with the powers? Hey. All right. So, make sure your mics are muted. Um, just let me just take notes of the time so that we could cut that out. I apologize for that. Right. So, everybody just take a deep breath, lock in. We want to make sure we focusing on the max and not anybody's else's behavior. All right. So, just making one last announcement that any outline that again you will be removed from the group. All right. So, let me just, you know, make sure I have the names to to to um remove. Right. Right. So, everybody, we are making sure to lock in. We are not going to let that disturb us. We need to make sure and have our brains fully on the mats. Right. That is all I'll say about that. So when we have indices of the same base and we are dividing we are going to subtract. So we took the seven minus the four and we got y to the power 3. Good. Anything raised to the power of zero is type in the chat. Anything raised to the power of 0 is one. Right. Right. So that's one. Anything anything I could take anything. I could take 11 and raise it to the power of zero. I'll still get one. And anything raised to the power of one you just put it back. So that's why we say 10 to the^ of one. That one there is understood. I could put 17 to the^ of one. That's still 17. Very nice number itself. A to the m raised again to the power of n. What we do with the powers is that we multiply it. x to the power of 2 raised to the power of 3. So that's x to the power of 2 * 36. So we have certain rules with roots in it. x to the power of m / n. So if I have x ^ of 2 over 3. This tree which is in the denominator goes here. This two which is in numerator goes here. So get we get the third root of x^2 the right cube root a to the power of m when I put that in the denominator the sign changes. So if I have x to the -2 putting it in the denominator I get x to the pos2. So if I have a b all raised to the power of n everything in those brackets is being squared. So question if I have 2x raised to the power of two what is that? The two is being squared to give 4. The x is being squared to give x^2. So that's right Elijah. 4x^2. What if I had division instead? What if I had 3 overx squared? The rule says that everything in the brackets is being squared. So your numerator is being squared. 3^2 is 9. Fix my line there. X is being squared. So 9x^2 is correct. So just making sure we understand the laws of indices. We're going to do a pass question. when it is you're solving linear equations, you're guaranteed to do that in the exam, right? So, yes, you should be seeing me on YouTube now. Right? So if I ask you, right, YouTube people, so you didn't miss anything. If I ask you to solve 2x + 5 is equal to 7. The first thing you want to do is between the two and the five, you're going to bring across that five. Good. So when you bring across the five, it's going to turn into min -5. You change the operation from + 5 to minus 5. 2x is equal to two. Take the two, put it across. It turns into division by two. So, x is equal to one, which a lot of people said. Very nice. So, let's just hit a pass question one time. Like I said, they tend to mix these questions with expansion. So, the first thing I want to do is expand 3 * r - 3 * 5. Bring all the hours on one side. So 3 R - R bring the -5 across - 15 turns into + 15. So you get 15 + 3. So 18 is equal to 2 R. Now you bring across the two. So you get 18 / 2. So the value of R everyone very nice Elijah it's 9. R= 9 right. So that's linear equations. You do need to know how to solve a pair of simultaneous equations. So you're looking at this is the example that we'll be doing. Simultaneous equations is where you solve for two unknowns usually x and y. 3x + 2 y = 12. So you could label them so you don't lose track of it. And 2x + 2 y is equal to 10. So the first question you ask yourself so yes the elimination is the best that's yeah we will be doing that at Angelica we doing everything so the elimination is in my opinion the shortest and it the ideas behind it a little better than the substitution so if it is that you take a look at the coefficients of x and y which one of them have the same the same coefficients The y they have a two in front of them. Now you take a look at the signs. Both of them have the same sign. So would we add the equations or subtract them? We would subtract the equations. If the signs are the same, we subtract. If the signs are different, we add them. So we subtract the equations. 3x - 2x = x. 2 y - y is 0. Right? So you want to make sure and put a little statement saying equation one minus equation two. So 3x - 2x is x. 2 y - 2 y is zero. It goes away. It's eliminated. Right. Um, is it Pman? 3x - 2x is x. 2 y - 2 y is 0. And then 12 - 10 is 2. So we get that x is 2. Very nice, Shaniah. So that's half of the work. Now you need to substitute x= 2 back into one of the equations. You could choose equation one or two. So substitute x + 2x 2 is 6. Right? So everybody here should be able to solve a linear equation. Bring it across. So 2 y is = 6. We get that y is 6 / 2. So y is 3. Right? That there is simultaneous equations. The next objective is to solve linear inequalities. Linear inequalities is the they're very similar, right? Is almost the same as normal linear equations except for the fact that it has an inequality sign. Right. Right. This is food, cake, and ice cream sort of sort of um questions. Uh multimaneous equations. Yeah, I just want to write. Oh, you can stack anything. All of this will be um in the notes. All right. So, a couple tips. Yeah, no problem. A couple tips is make sure you label the equations so that you keep track of it, right? when you write your statements, equation one minus equation two, they know what you're talking about. Nice. So, with um inequalities, there are two things you need to look out for. There are two situations where you need to change the direction of the inequality sign. The first situation is if you take the entire left hand and swap it with the entire right hand side. The next situation is if you multiply or divide by a negative number, right? And then if you look at past questions, they tend to ask you to represent your answer on a number line. So if I have the less than or greater than symbols, do I shade the circle? You don't shade the circle. I'll show you what I mean by shading. you don't share the circle. But if it's inclusive, less than or equal to, greater than or equal to, do you share the circle? Yes, you will share the circle. So to put that into perspective, let's just do a question. So if I tell you solve for x where x is a real number, you bring all the x's on one side. So 5x turns into - 5x bring all the numbers on the other side. 2 - 8 -x - 5x is chat -6 x very nice and 2 - 8 is -6 so now look at this I'm going to be dividing by a negative number so what happens is that I need to change the direction of the inequality sign so x is greater than or equal to positive 1. So yes, there was a sign change. So if I were to represent it on a number line, right? X greater than or equal to one. If I were to represent it on a number line, I put a circle over the one, right? Nice neat circle. Do I shade it? Yes, I do. It is inclusive. So x is greater than equal to I shared it greater than greater than or equal to one. So I put the arrow to the right. Yes. So yes you you change the direction of the sign when you are dividing by a negative number. Let's take a look at our next question. 3 - 2x greater than 5. So bringing across the tree it turns into minus 3. So -2x is greater than 2. I'm going to now divide nisha by a negative number. 2 / -2. What happens? This sign changes direction. So I get x is less than -1. If I were to represent that on a number line. No, that's not um necessary faith. If it is that it's not always that x the sign faces x. It could be less than greater than. It could be anything as long as you're making sure to follow the rules. So if I were to represent x is less than1. I'm going to put a circle over the one. Do I shade it? Lie. So you're saying whenever you get negative number you flip the inequality. Whenever I divide or multiply by a negative I flip the direction of the inequality sign. Correct. Okay. And because it's not inclusive I'm not I'm not shading it. And where do I point my arrow? I point it to the left. Right. X is less than1. So yes, this is subject of the formula, Angelica. We're making sure to do this. We need to take these equations and make R the subject. So for the first one, I have R - H is equal to RH. With these questions, right, everyone, the first thing I want to do is bring all the terms in R on one side. So bringing across the RH R - RH bring across this H, it turns into positive plus H. So factoring out the R, right? R is common to both terms. Factoring it out, R * 1 is R. R * H is RH. And then to make r the subject it'll be h all over 1us h. Think yall get that? Make sense? And what about part two? If I were to make r the subject for part two, v is equal to pi r 2h. So I bring the pi and the h. It's all trapped by multiplication. So when I bring it across, I change the operation to division. So I get v / pi h is equal to r². Am I finished? Let's go over the first one again. So for the first one, I bring across the RH. Does that make sense? I bring across all the terms in R on one side. Yeah. All the terms in R, I brought it on one side. Then I factored out the R. So Miss All right. You say you're bringing all the terms of with R on one side. to lock in everybody all the terms are on one side of the equation. So this have an R in it right and then RH that have an R in it. So I brought it on one side that I factored out the R. So all right you have to factor the RH it would not have divided saying that RH together is a multiplication sign. There is a multiplication sign. Yes, but both of these terms have an R in it. And I'm trying to make R the subject. I'm trying to isolate R by itself. So I factored it out and I get R * So when I factor it out, L * 1 is R. R L * H is RH. Factorizing. Factorizing is where you ask yourself two questions. What is common? What is missing? We're coming up to factorizing. And then you divide by the one minus h. So let me know if you're good with the first. Which line? This line where I factored it out. So do you see that there's an R in both terms? There's an there's an R in both terms with me there. There is an R, right? So that is common to both terms. I'm factoring it out. Right? So I factor out the R and I ask myself what is missing. R * 1 is R. R* H is RH. Right? So that's it. That's all. Okay. [Music] And so question like you just have to factoriize it. It depends on the it depends on the question if it requires you to factoriize it. So I'm going to make sure to go through factorization and there is um we look at these kinds of questions when we do functions as well. So we will come back to an idea behind this. So let's just move forward with part two everyone. I brought across the pi and the h I divide it. So volume is equal to pi over h which is equal to r 2. I'm not finished. I need to make r the subject. This one this one different now. This one is where you're moving the um multiplying by each. You kind of divide when you exactly have. So this one I change multiplication to division and Jonathan is correct. I need to square root it. Angelica square root of volume over pi h right to make r the subject right. So I'm making sure to do let's let's do the factorizing first. Right? This is objective 40. We need to know how to factoriize here. Um there are four types of factorization. We have basic difference between two squares. We also have factorizing four terms and the big one quadratic factorization. The question will not tell you which one to do. You have to recognize which one to do. So for basic factorization, right Rachel, everybody, you ask yourself two questions. What is common? What is missing? So this 4x and 12 y, what's common? And then when you see them two letters together or them two numbers together, you do yourself that it have a multiplication sign in it. When you're doing factorization, you know definitely you have to pull out which one is which one of the factorization you're using. When it comes to that one, it is they have a lot of letters. I understand it. It is be different. Don't worry, don't let the letters intimidate you. It's just like a different letter. No problem. So everybody between the four and the 12 we able to see that four is the highest common factor. X and 12 X and Y are different. So we are going to factor out of four. So 4 * X will give 4X. What's missing? 4 * 3 Y will give that 12 Y. Right? Then the next style is difference of two squares. You have to recognize that if something is in the form a^ 2 minus b^ 2 I could immediately factoriize into a + b by a minus b. So yeah um we call this like a dots dot situation. When you see x^2us 4 try again Jonathan when you see x^2 - 4 that is in the form of 2 x^ 2 - 2^ 2 so when I factoriize very nice it will be x - 2 by x + 2 right right lovely the next style is factorizing four terms so you're looking at expression and you're seeing four terms. The technique everyone is to first look at the first two terms 2x^2 and 8x and factor out what's common. So with 2x^2 and 8x we factoring out a 2x 2x will give 2x^2 2x 4 will give 8 x. So, I placed in the brackets x + 4. I take it up and I rest it down here. And I ask myself, what do I need to multiply by to get these two terms? Somebody tell me something. What do I multiply by to get -3x - 12? So, so making sure to tell me a sign with it. In this case, the sign will be negative. is -3x - 12. Last step, you bundle it up together, you're going to get 2x - 3 * x + 4. Very nice scenario, right? 2x - 3, right? So, yes, we're going to look at quadratic factorization now. So, quadratic factorization is Yeah. Go ahead. Um, you could just one more time. Um, which line? This line. So, what's common between the first two things? So, not just two, we also have x that's common. So, the entire thing we pull out is 2x. So, the 2x is 2x^2. 2x 4 is 8 x. And you just trust me on this. Whatever you put in the brackets, you take it up and rest it back down here. You just do that. Okay? And then you look to see what do I need to multiply by to get the next two terms. I'm going to multiply by -3. [Music] Yes. And the last step is to just bundle it up. So to answer the question, does it matter like the order? No. Because it's you could put x + 4 and then put 2xus 3. it will still be correct. So, we're looking at we look we will look at this in quadratic factorization as well. To do this, you need to know the AC method. So, drop a in the chat if you know how to factoriize a quadratic using the AC method. So, the first thing you have to do is play the factorizing game. Yeah, that's what we call it. Find out what the middle. So you have to set up the game. You need two numbers that multiply to give a c a c 2 by two four but add to give the b term five. So playing the game everybody what would those two numbers be? Four and one. So 4 * 1 is 4 and 4 + 1 is 5. So I have found my two numbers and now I am ready to to factoriize my quadratic. All right. So let me just make sure I write properly. + 5x + 2. So y'all can um ask the questions in the chat. I'm always reading the chat. Um, instead of 5x everyone, I'm going to put + 4x plus x. Right? I don't put the medium. I put 4x + x + 2. Right? So, m everybody, we're looking at the first two terms. And when you look at the first two terms, you're going to ask yourself, what is common between them? between 2x^2 and 4x 2x 2x. So I'm going to factor out the 2x. 2x is 2x^2. 2x 2 is 4x. So I'm going to rest down that x + 2 here and ask myself, what do I need to multiply by to get these two terms? That's right. For now, I need to multiply by + one. So I bundled it up 2x + 1 by x + 2. Right everybody? So you do need to know how to factoriize your quadratics. All right. Take it in. Take it in. You have to know how to factor a quadratic. What if it was just x? What if what was just x? Is it wrong to put one x here? No, it's not wrong. If the one is understood, so that's why I didn't put it. You could totally put one and it will be totally fine. Yes, x by x would give you x². So there is another um way of representing your quadratics. It's called complete the square form. Everybody knows how to complete the square. So there are several methods to complete the square. The most popular one is the formula method. Right? Luck in everybody. This is an important thing that can come in your question eight. So upon completing the square, your quadratic will now be in the form a into x + h all 2 + k. So we have two formulas to find h and k. Please note that your quadratic did not change. it is simply in another form right there are some other methods you know admat people will be on these kinds of methods right so what you want to do is memorize yeah these are your formulas you're memorizing h is equal to b / 2 a k= 4 a c minus b^ 4 a this is not in the formula sheet you have to memorize it Yeah. Right. So, maximum minimum roots with the formula. Yeah, I understand. So, okay, you have this is your example to do it. Right. This topic is called completing the square. You can use any method. The formula method is the best in my opinion for C segments. So this is your quadratic 2x^2 + 4x - 1. Type in the chat what is the value of a, b and c. a is two, b is four, c, you take the negative with you negative one. So I'm just asking for the coefficients now. I mean that's how that's the coefficients 2 41. So your formulas h is b over 2 a. You substitute b as 4 a as 2. So 4 / 4 is 1. So for the value of k the formula is 4 a c minus b ^ 2 all over 4 a. So like everything you write your formula you substitute a has 2 c has1 you're making sure to use brackets everybody 4 is 8 um -8 - 16 over 8. So being careful we have -4 over over 8 which is -3. I mean you could depending on how you write you could line it up side by side. You have to use both. You have to use h. You have to find both h and k. So everybody knowing the values of h and k tell me what is the answer? your final one. Well, you're starting off with a a is what? Two. Two open brackets. X + H. H is 1. So, X + 1 2 + K. K is -3. And that everyone, yes, that everyone is how you complete the square. All right. ABC method. Well, I don't know what you mean by ABC. If you want to do other methods, there's other methods in the notes. So, Ada, it is not B^2 minus 4 A that is I think you're remembering the quadratic formula, right? So, everybody ask your questions in the chat. I will not miss your question. I'm going over this when we do relations, functions, and graphs. I'm do I'm I'll do it over the way the the syllabus is is outlined. I have to do it over in relations, functions, and graphs. So, what it is everybody, the general idea of it, you write your formulas, you substitute, you find the values of H and K. You have to use both formulas. You have to know H. You have to know K. And your complete the square form is in the form a into x + h all^ squ + k. So there are other ways to do it which are much more complicated. It's in the notes but this is admat people will know this more. This is something called perfect square and it's very very integrated. And then this is the coefficients method which is way too long. Everybody here should be doing formula method, right? They give a formula sheet, but you have to know H and K. It's not in the formula sheet. You have to know it by heart. All right, let's do a um solving quadratic equations. To do this, you need to know the AC method. The first step is to set up the game. It's called the factorizing game. So, Brianna, everything that I'm doing will be uploaded right after the session. These notes will be sent to the group as well as the Google Drive. The AC method says I need two numbers that multiply to give A is 1, C is -42. And they must add to give the B term. The B term is one. So I want everybody to play the game and tell me the two numbers. Play the game. Two numbers that multiply to give - 42 but add to give one. So check back your signs. Mark Jude 7 and - 6. So 7 * -6 very nice Rachel will give -42 7 + -6 will give positive one right so you don't write the b term you instead write 7 x - 6 x - 42 is equal to 0. So you now have four terms. You look at the first two terms. What is common? X. What is missing? X by X is X2. X + 7 7X. Take up what's in the brackets. Put it down here. What do you need to multiply by to get these two terms? What goes in that box? Not just six. What's else? Exactly. Right everybody when you're telling me the sign right so x - 6 by x + 7 is equal to zero. So what would be the two values of x? What would be the two values of x? Right? So everybody could use the chat to um to write your answers to to do um to write anything you want in the chat. Right. X we have either X - 6 is 0 or X + 7 is 0. So X is equal to pos6 X is equal to -7. So your two values of X are 6 and -7. There are instances, yes everybody, recently recent changes, there are instances where you need to use the quadratic formula. We need to Yeah, go ahead. It's likely to get it's likely for the answer to Z when you substitute the x into the um equation, right? So when you substitute six inside here it will equate to zero. Yes everybody there are instances and very recently last year you had to use the quadratic formula. It was last tested in June 2024. Normally they don't. If you look at the past papers they don't bring it but now they do. All right. Um I just finish and help everybody before we go into pass. you need to be able to solve for we have it in the part of a question that I'm doing right so everybody could use the chat to talk to make sure that um it the recording is recorded properly so I'm doing I'll do that in the part of a question you have four mangoes and two pairs costing $24 while two mangoes and three pairs cost $16. So I want you all to turn this into algebraic equations. Let X be the cost of a mango and Y be the cost of a pair or you can use M and P show. So four mangoes and two pairs cost $24. Whereas two mangoes and three pears cost $16. All right. So with a pair of simultaneous equations, you could go ahead and solve it to find the cost of a mango and the cost of a pair. Right? So we need to be able to solve. Let me just get the question. I I mistakenly erased the question in very rare instances, but it can happen. I think the last time it happened was 2018 where you get a quadratic and then the other equation is linear. So for example, if I have the quadratic y = x^2 + x - 4 and then I have the linear equation y = 2x + 2. What would you do to solve right? We have y equals this and y equals that. So, I'm going to equate them, right? I'm going to equate x^2 + x - 4 to be equal to 2x + 2. Yeah, very nice out. Then you bring all the terms on one side. Bring the plus two across. It turns into minus2. Then I simplify. Everybody's okay. Then I simplify. When you look at this, you'll see that it's a it has a x². So that's a quadratic. This is of the form y = mx + c. So that's linear. So again, we get a quadratic equals z. So everybody, you need to know how to factoriize this quadratic, right? So all I did was equate, bring all the terms on one side and then I simplified. So I need everybody to be paying attention to how we do this. Are you up to this point where I equated? Drop a line in the chat if you're good with the equating part. Nice. Rachel, I just where I get the sum, I made that up and put it down on the board. Right now, everybody, the next step is to take these terms and bring it across the equal sign. 2x turned into - 2x + 2 turned into -2. And I'm doing I'm going it over. I'm I bought it, of course. Good. Now I simplify x - 2x - x -4 - 2 is -6. So what we have here everybody is a quadratic. Good. All right. That's the quadratic. Nice. How do we factoriize a quadratic? Exactly, Rachel. I wanted to get zero on the other side. We need to use the AC method. In the AC method, the first thing you do is you play the game. So, we need two numbers that multiply to give AC, but add to give the B term -1. So, everybody tell me the two numbers. Yes, Catherine. So, this is what you do when you have a linear and a quadratic. Try again. -3 and positive2. That's right. That's all. -3 * 2 is -6. -3 + 2 is -1. So I put -3x + 2x. Factor out what is common. X by X is X^2. X by - 3 is - 3X. Put down back the minus 3. What do I need to multiply by? That's right. Two. So X + 2 by X - 3. What are the two values of X? because it's a quadratic and on a straight line. So you have two values of x. So you have either x + 2 is 0. So x is -2 or x - 3 is 0. So x is 3. So you got two x values. Now you need to find the corresponding y values. So, you're going to substitute it into either equation. If I were you, I would go with the linear equation. What would you get? 2 * -2 + 2 is how much? -4 + 2 is -2. And when x is [Music] pos3, 2 * 3 + 2 is 6 + 2, which is so yes, you can substitute into the either one, but the easier one is the linear equation. So everybody, you have two pairs. So you have x is -2, y is -2, and then you have a next pair. X is 3, y is 8. Right? So this one, it's not very common, Liam, where you're going to get a linear and a nonlinear. you more get a pair of simult you more get more get that. So these questions I think the last time was 2018 a linear and a nonlinear. Right. So we're coming up to the last couple of objectives here everybody. Have you all ever seen this before? Hope it stays in Cxc headquarters. I hope so too. Nella that's right Goldie. If it is that you you could write it as points. A point is in the form x comma y. So yall have seen this before, right? If you know this, you can immediately write down when you expand a when you expand it, you immediately write it down off this form. So if it is that I had x + 2 all squared and I told you to expand. Very nice Elijah. is like perfect squares. If I told you to expand x + 2 all 2 following the rule, it will be x 2 * x * 2 is 4 x + 2 which is 4. Sorry, I'm following this form. So, a 2 that will be x 2 a 2 by x by 2 4x and then what's 2^ 2 that's four. So David, that formula is the quadratic formula and we will look at a question momentarily on that. So the second one is a minus b all squ which is it have a minus in it. So you have x a 2 - 2 a + b 2. And the last one, what does that look like? A minus B by A + B. What does that look like? That is your very well-known That is your very well-known difference of two squares. Okay. So, let me just make sure everybody could do they could do this, right? It's very simple to follow a form and write down the answer. So I need some lines. Hold on. I need some lines to write on. Good. So the form is a + b all 2 is equal to a 2 + 2 a + b 2. So you're learning that off. And I'm telling you I want to find x + 3 all squared. So a is x. Everybody following a is x. So a 2 will be x 2 * x. b is 3. B 2 is 3 2. So I get x^2 + 6 x + 9. If you know it really good, you could just skip this step or you do it in your mind. Does it come frequent? It could come. It could come when you expand it. When you expand a binomial, you will get this being b. That's just how it works. That's just how it works. I have a plus here. So, plus three squared. Nice. The next set of objectives is about proportionality. So, you could have direct proportionality, indirect proportionality. So, I just want to snap something. This is the shorter version instead of always having to expand it. Right? So for direct proportionality right everybody lock in yes you have to simplify always for anything you have to simplify. So if y is directly proportional to x how do I write that symbolically y is directly proportional to x this is the symbol that I use for proportion right well right it's like a alpha y proportional to x but we don't like to we don't work with that sign we will change it to an equal sign when you change it to an equal sign you have to introduce the constant key So we have y = kx. So this question says y varies directly as x. Calate the values of t and u. y varies directly with x. Removing the proportionality sign I get y is equal to kx. We have enough information to find k. We substitute x as 6, y as 3. So what's the value of K? 3 / 6 which is K is a half. Very nice Elijah. So our equation everybody is Y is equal to a half X. So when X is 10, can you find the value of Y? It will be a half of 10 which is five. Nice. And using the same equation when y is 9, can you find the value of t? you um you can multiply by two throughout. So 9 * 2 is t. So the value of t is 18. Everybody good with the 18, right? It's 18. And what is the opposite of uh you when you bring across well you multiply by two so 9 is 18 I'm multiplying by two throughout to get rid of the 1/2 I multiply by two so 9 2 is 18 a half by two is one. So one T. All right. So I'll do it another way in case this does not registering. So when I bring across the 1/2, I have to divide by a half. Correct? 9 / a half is t. So instead of dividing by a half, we change division to multiplication and invert the fraction after it. So 9 choose 18. All right, the next style is indirect proportionality. So indirect proportionality y is indirectly proportional to 1 /x. So so we don't use this proportionality sign. We instead we change it to an equal sign and introduce the constant key. So that came as recent as January couple February, March, April, three months ago, four months ago. We are given that y is inversely proportional to x - 2, right? So it will be 1 over x - 2. It's inverse. So I have to do 1 over xus 2. We don't work with the proportionality sign. Instead, we change it to an equal sign and introduce the constant key. Everybody with me? We don't work with proportionality signs. We we replace it with an equal sign and introduce the constant key. From the question, I'm given x= 9, x = 11, y = 9. So, I substitute and I'll get 9= K over 11 - 2. So working out the value of K, what would you get? 9 * 9 is 81 which is equal to key. So the equation that you're working with right replacing K you found it 81 very nice you found it to be 81. So this is now your equation. The question said find the value of y when x is 29. So you substitute x as 29. y is equal to 81 over 29 - 2. And I'm seeing correct answers in the chat. 81 / 27 is three. And that is how proportionality comes right. So yeah, direct and indirect proportionality. So when you take a look at the recent past papers, this is the trend that you're seeing, right? Those those who are um everybody focus are those who have been doing past papers have you recognized that the entire question to is on algebra but algebra is something you use throughout the whole paper right algebra is foundation so it's used throughout the entire paper one of the noticeable trends is that they're incorporating a measurement idea in the question you all saw that And then another trend is where you have to recognize you could factoriize numerator and the denominator and simplify as another noticeable trend. So what we'll do is um watch a pass question for you to see that. So if you take a look at last year, just last year, even even the year before, they keep putting a measurement idea in it. Like they had the measurement man creating question two, right? So this is the seedbed. So everybody, I have eight minutes to do the question. So y'all need to lock in. Laura needs to put mesh around two seed beds to protect her seedlings. All together she uses 60 m of mesh. Lock in read the question one of the seed beds is a rectangle. You see again that the other is a square right? That's a square. The width of the rectangular seabed is x m. we see in that the length of the rectangular seed bed is three times the width. So what would it be in terms of x? What's the length? Three three times its width. So that's right. This is 3x. The length of a side of square seed is y m. Using the information below, derive an expression for y in terms of x. So they gave us some information about um putting mesh around. When you hear this word, what comes to mind? Around around the shape. Exactly. Liam perimeter. So what is the perimeter of a rectangle? Two multip. Nice. We just figured out the length and the width was given as x. So combining 3 + 1 4x 2 * 4x is 8 x that is the perimeter of the rectangle. What is the perimeter of a square? 4 * s. The side here is y. So 4 * y 4 y. We were told that the entire perimeter is 60 m. So I'm going to take the 8x plus the 4 y and get 60 m of mesh. We want to derive an expression for y. We need to make y the subject. So bringing across the 8x I get 4 y is = 60 - 8x. Yeah. And now I could what else can I do? Elijah tiana there's a four. There's a 60. There's a eight. I could reduce it. You can leave it up to here. There's only two marks. Or you could divide by four. 60 / 4 is 15. 8 / 4 2x. So y is = 15 - 2x. Nice. Now we get into the quadratic everybody. But 4 y = 60 would be correct, right? Yeah, that would be correct. All right. But reading over the question. Ah, I read over the question there. What did they say? They said simplified. So actually you needed to have do done it. You have to reduce it because the question specifically said so. So you wanted 4 y in terms of x. So to yes the answer is yes you need needed to have simplified it for this particular question. All right. So I'm going forward Haley um it was just to find the perimeters add it and then simplify it. The area of the rectangular seedbed is equal to the area of the square seedbed. Everything was divided by 4. 60 / 4 is 15. 8 / 4 is two. Right? So that is the expression there. Yeah. It's all over the four. All of this is four. Divided by four. Everything was divided by four. But sometimes in certain I guess in certain according to what type of thing you're doing in math when you have questions like you don't have to leave it as it is you can break down this 60 / 4 is 15 and then 8x / 4 is 2x so that's what happened I say this this particular equation here whatever this is here what the name of it trying to explain that you can divide it down there like this. But in certain area in math you can't divide nothing. You have to leave the answer as it is. So yeah it depends on it depends. It depends on what you have in like if you're talking about like algebraic fractions like we had x + 4 / 24. You cannot um reduce that. You have to leave it just like that. All right everybody. So going forward we need to deal with area. What is the area of a rectangle? by right length by right length by width length by bread. So we have 3x * x which is 3x^2. What is the area of a square? Oh gosh looking like a square but area of a square is side by side which is y by y which is y 2. So, we have area of the rectangular seedbed is equal to the area of the square seed. So, I'm going to have to equate it, correct? 3x^2 is equal to y^2 cuz they said it was equal. I need to use the information from part a and show this from part a. Don't I have an expression for y? Yes, y is 15 - 2x. Right now, this is where I need y'all to lock in. This is in the form a minus b^ 2. Right? It's in the form a - b^ 2. Agreed? So when I expand it, it is a^ 2 - 2 a + b^ 2. That is what you're learning of. So what is a formula? Yeah, it's like a formula. Okay, sure. Let's do that. So 15 - 2x * 15 - 2x, right? So those who know how to use the formula, you could go ahead. We're going to expand. 15 by 15 is how much? That's 225. 15 by -2x is -30x. -2x 15 is -30x. And -2x -2x is pos4x^2. So now you simplify -30 - 30 -60x bring across the 3x^ 2. It turns into - 3x^2 and 4 - 3 is 1, right? 4 - 3 is 1 x^2. And isn't that what we wanted? We wanted x^2 - 60x + 225. So when you go up here, that is exactly what you wanted. Yes. So when you show something, what do you write? You write qed for shown. What mean again? It's Latin for quadrata diamond stratum. But you don't have to know all of that. It's Latin. Q means shown. Now I want everybody to take a look at that quadratic equation. We just don't have to. You should you show something. It's very satisfying to to write QED. You can write a box. You could write the two dashes, right? I'm not sure I understand the question. You can type it. I'm not sure I understand the question where you just said you had a formula. Um, you could do this do it in this way instead of formula. Yeah. Expanding. Yeah. Cuz I know some of y'all not too sure about the form. It's like a formula. Yeah. Formula form. So you could expand it. You show it and you write QED. Yeah. Go ahead. Okay. But you could 16 - 2x itself. You couldn't do that. What? Multiply by itself. 15 - 2x multiply by itself. Yeah. No problem. Jonathan sir 15 - 2x multiply by itself. 15 - 2x. Yeah. All right, everybody. Yeah, go ahead. If you could use it on your calculator. So, you can solve quadratic equations on your calculator. That's not a problem. But you want to make sure for part two that you are showing you're working with the quadratic formula. So, everybody, you have to use the quadratic formula. What is the quadratic formula? It's in formula sheet. That is your quadratic formula. So looking at your quadratic, can you tell me a, b, and c? a is 1, b is. You take the negative with you. -60 and c is 225. So don't worry about the numbers. The calculator will do it. So you have the root of b ^ 2 - 4 * a * c all over 2 * 1. us the root of 3600US 900 using a calculator. So you have 60 plus or minus the root of 2700 all over 2. It said to solve. So you're finding x the roots. So everybody you have either x is 60us the roo<unk> of 2700 / 2 or you have x is = 60 + the roo<unk> of 2700 over 2. So using your calculator, you could get the decimals. It's in the schedule. Yeah. Once you know the formula, and even if you don't know it, it's in the formula sheet. 60us the root of 2700 all over two. So as a decimal, that's approximately 4.019. 019. Everybody got your answers? 4.019. And what's the other one? You have to change it to a plus. Yeah, but it's not it's not advised that you approximate early in the question. So, you want to just use the Casio and get put the whole thing inside here. You get 55.98 55.98. Everything was in meters, by the way. You know, it's like measurement. You have to put your units to four significant figures. All right, everybody. So, 55.98. Yeah, we'll just um use the four significant figures that is solvent and significant fig. So 55.98 you significant figures are non zero non-zero digits. 55 5.98 this zero here because it's less than five we don't do any we don't round up we just leave it as 0.98 so write it in pen diagrams and graph sign pencil so this is the last piece here everybody the last piece we a little over the time last piece calculate the area of the two seed beds you know so they was doing the most for this um for this question cuz this is a long question. It's only two marks. But we're going to do Yes, it's all to two to three. I just made it to four. So, we have two values of x. We know that y is equal to 15 - 2x, right? We had solved that y is equal to 15 - 2x. So, would you choose the 4.019 019 or will you choose the 55.98? Which one if you were to choose? Does it matter? It matters because if I choose this right, if I choose 55.98, what will be the value of y? When you solve that, you'll get a negative value. It will be a negative value. And does does a length be negative? Is a length negative? A length cannot be negative. So since lengths cannot be negative, then we need to choose the smaller value. X is equal to 4.019 m. So, we're going to choose the smaller one, right? Because X or Y cannot be negative. So, the 55.98, it's too big. So, we're choosing the smaller one. X has 4.019. Exactly. So, substituting when X is equal to 4.019, 01 9 y is 15 - 2x. So what's the value of y 6 6.962? Yes. All of everybody everything that I write everything that I say everything that I snap onto the board is in the notes you don't have to worry. So we have the value of x we have the value of y. So could we go ahead and find area? So the area of the rectangular seabed, we had figured it out. Chat people, make sure you're doing the maths. Area of the rectangular seed bed is 3x, which is 3x^ 2. So that's 3 * 6.962. So how much is it? So Zach is making sure and ka as well making sure to give my answer 96.9. This is measurement. So we have me squared area. This will also be the area of the square bed cuz we were told in the question the areas are equal. So both of them has have an area of 96.9 m squared. This is area. So it's me squared. So, make sure to put your meter squared. So, is everybody okay with that? It's 2:40 and we're supposed to finish 230. So, we just went over by 10 minutes, but that was okay. I know it's um I know it's I have relations, functions, and graphs. So, that's why I I I should have put the extra 15 minutes here. Anyways, so according to the schedule, we're going to dive right into the relations, functions, and graphs. Oh, are they giving my wrong answers? Are they giving my wrong answers? Hey, who give me the sub 96.9? Thank you to the people who give me my correct answers. Oh, that was the fun from before. Y'all give me the wrong thing. Let me let me fix it. So, the correct thing is 145.4 me squared. 145.4 t test. So, you don't want that to happen to you on your exam. So the schedule is in the group. Um so 2 hours and 15 minutes 2:45 straight until 4:45. So Nisha if you have a Casio it'll be really easy but if you have a sharp you have to put brackets brackets. Brackets saves lives. No hate to the sharp. I grew up with the shop but the Casio is the best. You want to put the 27. It's root of 2700 and nisha root of 2700 if the shark better. Well, Elijah, I don't know about that, man. All right, everyone. So now according to the schedule we go directly into relations, functions, graphs. So I need you to take a deep breath. We have two hours of this. I could give y'all the could give you all 2 minute breather till 2:45. The notes will be posted right after the session. Everyone, two minutes to stretch. Y'all see the time on the board? Two minutes. Right. All right. So, I just want to say with a resounding yes, you will get all of the notes, after class, after the session, you will get all of the notes that you see here. All 300 pages. Oh, by the way, if anybody has the paper tree from yesterday, they can make sure, you know, share it, send it. I didn't get it. All right, everyone. So now we are diving into relations, functions and graphs that comes in question eight. Functions come in question four and can show up in question eight. Right? So everybody seeing the screen ready to get started. What is a relation? So in order to define a relation we need three things. We need two sets of data. A rule between them and a direction of the rule. Right? So chat people lock in talking to the chat. Just making sure my Google Meet is Right. I'm just making sure my Google meet working there. So, just drop a while in the chat if it is you guys are still seeing me. I just making sure everybody So just a a quick announcement about the Google Drive. The operations team will grant your request to the Google Drive by the end of the day. They do so very quickly by the So make sure request access. They'll do so by the end of the day. Yes. Um, it's pulling from two different things. So, the Google Drive, the Google Drive link is in your WhatsApp group description. So, everyone, let's take a look at this question. Um, it's wait for my camera to come back on. But while it does just lock in, you're seeing two sets of data on the board. Can you You've been around numbers all your life. Can you tell me the rule between them? How do I get from 1 to 6, 2 to 12, 3 to 18? How do I do that? So yes, it's a one to one relation. Yes. But how do I do that? I multiply the value of x by six. So to write down the rule 12 has it y is equal to 6x. Agreed? Multiplying by 6. You see that? So this set of data is called your input. This set of data is called your output. There are four types of mappings. Can you start to name them? So the first type is this is section seven relations, functions and graphs according to the syllabus. So we have a onetoone mapping where one input is being mapped on to one output. So one piece of pizza for everybody. One to many where one input is being mapped on to two outputs. So one person getting two slices of pizza there. Many to one. So two inputs is being mapped onto one output. And then you have a many to many. Many inputs being mapped onto one output and then one input being mapped onto many output. So that's a many to many situation. Am I stuck? Okay, I'm good. So, you need to be able to represent a relation in various ways. The first one is an arrow diagram. So, you have four being mapped onto 10. So, that four is going to follow the rule when x is 4. Following the rule, 4 * 4 - 6, 16 - 6 is 10. Right? That's your output. So you do that for all the inputs to find your outputs. I'm sticking no. All right. So y is equal to 4 * 4 16. 16 - 6 10. Yeah, it has a couple multiple choice questions with this. Sometimes they also ask to determine the rule. So if I ask you you to put this in ordered pairs that's like a coordinate. It's in the form x comma y. So it's a set. So you open up curly brackets. It's a set. So you have the first point 4 comma 10 5 14 6 18 22 right so x comma y that's an ordered pair and I use curly brackets to put it into a that. All right. So, let me just make sure my system is good. Everybody just hold on cuz I think the YouTube people YouTube people is seeing me, right? All right. Nice. So, everybody's good there. ordered pairs. Out of the four relations, which of them are functions? You have two of them that are functions. One to one, many to one. What's the acronym to remember this? So, some students remember it by moo many to one and then one to one. Right? So one one and many to one those are functions. The other two of them are not functions. If you put normal brackets Angelica that is saying something else everything in maths have symbols and to represent a set you want to use curly brackets right Mr. Slinger went sets you want to use curly brackets. It means something else if you put the normal. So let's make sure we understand the concept of a function. I'm taking the input two putting it inside of my add tree factory. Does it make sense that it will output five? Because 2 + 3 is five. Nice. So how would I write it as a function? I take an input let's call it x. go through the function and it produces x + 3. So if I input two, it produces 2 + 3, the output is five. So that is the concept of a function. Yeah. 2 + 3 5 using functional notation. How many of y'all have seen this before? Right. So if I I um CX brings it like that as well. And if you don't like to see it like that, you could write it over how you prefer to see it. So what it says is that f is such that x is being mapped onto x^2. But that kind of weird. So we like to write it over f ofx is equal to x2. When you see this symbol, that means the inverse function. This is a composite function where g is being substituted inside of f. We're going to do questions so you could see it. How do you distinguish between a relation and a function? Well, we just memorized that many to one and one to one functions one to one relations are functions. But from a graphical perspective, what's the name of the test? It's on the board. It's called a vertical line test. So, you're testing the function that you a vertical line. If the vertical line, you all seen this before, a vertical line, you're using a vertical line drawing it through the function. If it cuts the graph at most once then yeah you have a function. If it cuts more than once then no it's not a function. So if I have the function x + 3 is that a function? This is the line x + 3. Is that a function? No. A vertical line. You know everybody if I if I take a vertical line any vertical line you could think of and I pass it through any part of the function it will cut at most once. So yes this is a function. It is yes. What about a quadratic? If you put a vertical line here you put a vertical line here. Wherever you put a vertical line, it will cut only once. So yes, it is a function. So it's a there's a very popular multiplechoice question about it. Which of the following represents a function? A B C D. Yes. Well, linear function, quadratic function. Um, this cuts twice. So, no. This cuts once. Yes. This cuts twice. No. This cut twice. No. So, the answer is B. Nice. If I ask you to draw a horizontal line, drawing lines. Everybody could draw lines. you've picked up your graph paper and drew lines before. So, horizontal lines, you know, horizon horizontal are in the form y= k where k is a constant. It's a number. So, if I ask you to draw the line y = 2, y = 0, y= 5. All of those muel are horizontal lines. So this is your xy plane. y = 2. It's a horizontal line. It passes through y = 2. y = 0. Oh gosh, you're looking so bent. You're taking your ruler. Don't forget your ruler. Why? Yeah. Go ahead. Take a repeat. That horizontal Oh, to carry a ruler. Make sure to carry a ruler and horizontal lines. So, y = 2 passes through y = 2. y equals 0 passes through when y is zero here. So when it says that you write a coordinate, this is 1 0. All the y coordinates are zero. This is the point 3 0 4 0. All the y coordinates are zero. Oh, so all of these anything off the form y= k it's a horizontal line. y =5. All of them will have a ycoordinate of5. So taking a look at the graph, what's another name for y = 0? What line is that? What what is that y = 0? The x axis. The line y = 0 not x is the x axis. y = 0 is the x axis. So those are horizontal lines. What if I ask you to draw vertical lines top to bottom? It is in the form x= k. k is a number. So 3 0 -4. So if I ask you to draw x= 3, you're looking for three on the x axis and drawing a straight line. x= 3. x= 0 is the green one. x= -4. Right? All of those are vertical lines. And looking at the graph, x= 0 is the same as what? It's the same thing as the y ais. Very nice. Right. So that's another name for the y axis. x equals z. And now you have the general lines. So to do general lines, you do need to find the intercepts. So lines are in the general form. y = ax + b. Some persons know it as y = mx + c. Right. Right. Another way could present themselves is in the form ax + b y is equal to c. a, b, and c are all constants. So to plot these lines, you need to first find the intercepts. You have x intercepts, y intercept. To find the y intercept, you're going to let x be zero. To find the x intercept, you're going to let y be zero. So, let's do a question. If I have the line x + 2 y = 6, find the y intercept. You're going to have to let x be zero. So 0 + 2 y is 6. So what's the value of y? Very nice. It's three. And how would I write it as a coordinate? Because that's a multiple choice question. How would I write it as a coordinate? A coordinate is in the form x comma y. So, x is 0, y is three. Very nice. 03. I also need to find the x intercept. You're going to have to let y be zero. Substitute in x + 2 * 0 = 6. x is equal to Afternoon. X is equal to 6. So what is the coordinate? You have to read your question properly and see if they said coordinate. When they say coordinate, you have to put it in the form. Don't forget your brackets. Write the um 6, 0 in brackets. Put some close on your coordinates. So you have the point 03 6. You plot it. Everybody could plot points. 03 6 0. Take your ruler, draw a straight line through it. And that right everybody seen the line. That is your line. X + 2 Y is equal to six. So that is how you draw the other general lines. The next objective is gradient. What is another word for gradient? It's rarely used but it can show up. Another word for gradient is slope. Gradient speaks to the steepness of the line. Yes, everybody. No problem. Those who just came in, gradient speaks to the steepness of the line. And what is the formula for gradient that you typically use in your questions? You have to know it by heart. It is not in your formula sheet. If I log out, I'll come back in. Just restarting the All right. Sometimes the Microsoft whiteboard it pulls a lot. So, so I have to restart it because I put too many pages inside of the board. Anyways, so you guys are correct. Another word for gradient is slope and it speaks to the steepness of the line. You have to know the formula by hat. So, y2 - y1 over x2 - x1. So I just making sure that I accepted everybody in the class and you're going to help me figure out this able. All right, just give me a second. Just give me one second. Um, a lot of people trying to come in and I just want to make sure everybody is inside. All right. Normally what we do sometimes is actually have two separate Google Meets. So, but anyways, it doesn't matter. It will work. Right. So just by looking at the line, am I sticking? Just by looking at the line, you can tell whether it's positive, negative, zero, or infinite. So I want you to put into the chart looking at it for the first one. If I have a positive gradient, will that be the first, second, third, or fourth representation? Right? Everybody answer in the chat first, second, third or fourth. So you will choose the line that is sloping upwards from left to right. First one. Nice. What if I had a negative gradient? Which one would you choose? You would choose the line that is sloping downwards from left to right. If I have a gradient that is zero, that is represented by which one? The gradient a gradient of zero is represented by the fourth line a a horizontal line right everybody gradient of zero is like you walking on flat ground you're not walking up the hill down the hill you walking on flat ground there's no gradient is zero and if it is that you have infinite where you undefined gradient that is represented by a vertical line. Right? So just by looking at the lines you are able to tell you are able to tell characteristics about the gradient. So a couple formulas that you need to know. The equation of a straight line is represented by y = mx + c. So what does m represent? m represents your gradient. So in this case m is 2. And the value of c represents your y intercept. Your y intercept is where it cuts the y ais. And when it is you're doing questions, you find yourself using this formula. y - y1 = m into x - x1. Very popular formula. It's called point slope form. We're going to do a question with that. You don't know about this. So, x1 y1 that really I mean there are ways you could find yourself not using it, but you should you should know it. It's it will cut your working by a little bit. So x1 y1 that is your point and m is your gradient. So when you substitute and you work your way forward you find the equation of the line one time. Yeah there's this formula is not like a new new formula if you look at it really closely. It is your gradient formula. What do you guys know about parallel lines? We'll do a question on it. What do you guys know about parallel lines? Parallel lines go on and on. They never touch each other. It's a sad love story. They never they never met in life. And we know that the gradients are equal. M1 is equal to m_sub_2. They have the same gradient. Perpendicular lines. What do we know about them? They meet each other at right angles or 90°. This is also a depressive story where they meet each other once and never meet each other again. And what do you know about their gradients? Their gradients are right inverse and flip. So negative reciprocal m_sub_2 is equal to1 / m1. So, and when you multiply it, you're going to get one. Lovely. So, you'll know your things. So, let's um Oh, a couple more formulas. So, if the question says length, distance, right? All of that. It means the distance formula. Here you go. The midpoint formula. Here you go. Yeah. Go ahead. Question. So this will show up in your question four. This year the entire question was functions but last year the entire question was corn geometry. Okay no problem. So is it oh no this year this year was coordinate geometry. So January 2025 was the entire question question four was on coordinate geometry. So let's do it right. We have a line segment. Segment means you know a piece of a line joins the point56 and 72. So I want everybody to use the midpoint formula and tell me the midpoint of CD. You add up all your x's. divide by two. Add up all the y's. Divide by two. And there's a comma between them. So substitute and tell me the answer. So to make sure that you substitute properly, this is still cack math. This is x1 y1 x2 y2. So you substitute -5 + 7 / 2 6 + 2 / 2. Answers in the chat. So 2ide by two. Elijah just check it back. 2 / by two. 1 4 miss. Yes. You can let me know what line is giving you trouble so I could specifically clear up that line from the second substituting. So substituting you're taking your x's -5 + 7 / 2 6 + 2 / 2. You're making sure to show substitution line. So I see a negative one popping up. You just check it back. 7 - 5 is two. So the midpoint everybody is one four. Make sure to put your brackets but you want to make sure and put your brackets for the exam when you're writing. What's your gradient formula? Sorry to the YouTube people. I see you on Just a second. My mic is is um taking its time to to come back on. So Rihanna has it. Y2 - Y1 over X2 - X1. Okay, I realize it's something is Oh, I'll type one. [Music] So, you should be able to hear me now. I just representing the screen YouTube people YouTube is running fine. So I got a little one minute break. So everybody should be doing the questions why it is that we have the little little things to sort out, right? It's because it have but my system is a standard hub system so it should handle everything but normally what we do is we we had separated it into two separate Google meets and and things like that right so everyone when it formulas you need to make sure and substitute properly your line your line segment have 56 6 72. Those are your two points. -5 6 and 72. So you could take your little pencil, put x1, y1, x2, y2. So you're making sure to substitute properly. The first step is to always write down your formula. y2 - y1 over x2 - x1. Substitute 2 - 6 / 7 - -5. Did anybody put their answer in the chat? 2 - 6 is -4. 7 - -5 is 12. So when we reduce we get a3. So the gradient of line segment CD is a3. Nice. The question went on and said find the equation of the perpendicular bis sector. That's just the perpendicular line. So the gradient of line C D we just found it to be negative a third. So what is the gradient of the perpendicular line knowing that they're negative reciprocals. Very nice Elijah. It's three right everybody. If I take this number and this number and I multiply it I'll get1. So I'm going to substitute the gradient to be three and the point right perpendicular bis sector. So that's going to cut at the midpoint which we just found to be 1 4. And this is where the formula comes in to those who have not seen it before. Y - Y1 = M into X - X1. You're going to use that formula. Okay, Elijah, you could put do it like that, but you have to find the value of C by substituting the point. So, y - y1 the tree is the negative reciprocal of a third. Yeah. So we substitute y1 to be four, the gradient to be three and x1 is 1. You can't leave your answer like that. We expand we get 3x - 3. Bring across the minus 4 it turns into + 4. So y is = 3x + 1. And that there is the equation of the perpendicular bis sector. So that's why the formula is much better. It it nice and succinct. The next last part you have another line. Another line AB is parallel to CD and passes through the point 01. Write down the equation of the line AB. So what do you know about parallel lines again? They never meet. They never meet. And what do you know about the gradients of parallel lines? They are the same. So if the gradient of CD is negative a third, then the parallel line AB, it's also negative a third. And then we have the point 01. When you see that point, that point that looks very interesting. 01, x is zero, y is one. So, what can you say? Um, no. Well, no. The the point cuts the yaxis at one. So you can say that the y intercept c is one. Does that make sense? When x is zero, you have the y intercept. So c is one. So you have m, you have c. So you can just write down that the equation is 3x. Am I stuck? Right. 3x + 1. I'm not saying the negative. It's negative 3x + one. Right ka. That's correct. So is where we get one from the one is taken from the point. The point 01 suggests to you that the y intercept is one. So this year the the entire question came on coordinate geometry. No. So your equation have to be in terms of y and x. So you don't need to substitute anything. y is equal they substituted m to begative a3 and c to be one. So that's it. That's all you had to do there. All right. So we're going to look at some work on functions. So graphically you could plot a pair of simultaneous equations. How would you find the solution? So if I plot 4 - 5x and 3x - 4 to find the solution, you are going to be looking at the point of intersection. Yes, you're going to look at the point of intersection. In this case, the point of intersection is 1. So that means that x is one, y is1. So that's as if it comes graphically. The next objective is to represent linear inequalities. So the first type is set notation. So values of x between -1 and one. What if I wanted to represent this inequality on a number line? values of x that are greater than or equal to -4 but less than 3. So people in the chat help me do this. All right, help me do this. X has to be greater than or equal to -4. Am I going to shade the circle? Yes. Very nice. Yes, it's inclusive of -4. What about the circle over the tree? No, cuz it's less than, right? It's you don't shade the circle there. So, it's trapped between there. So, that Oh, you just draw a line between it. So, that is between values of x -4 less than equal to x less than 3. Graphically, if I wanted x less than 4, well, you know x= 4 is a vertical line. And where would you shade? To the left or to the right? You will definitely shade to the left because you want values of x that are less than four, right? Ka, you you share to the left. Less than four. So these things help with linear programming, right? Going going to that now. In linear programming, part of your working is to identify the region that satisfy whatever your given inequalities are. So I have two inequalities x + y less than or equal to 5. So the first thing you're going to do is draw the line. You find your x intercept, your y intercept, and you draw the line. If I want less than or equal to five, is that below or above the line? Less than or equal to five? That's below the line. Y is less than or equal to X. Is that below or above? So this is the line y = x. Am I shading below or above? I shade below. And where the region intersects that this is the region that intersects. So that is the region that you are shading to satisfy both inequalities. So that is just working. You really just shade the green part. Now you don't have time to shade in the exam. All you are doing is taking your ruler and drawing This you don't have time to be coloring. This is not this is this is not um time to color. Mr. Springer will do over like in the geometry section. The objectives are in the geometry section. So he'll touch on it back in geometry. Ge geometry have that transformation thing. I'm focusing on making sure I finish this cuz I already went over by like 15 minutes. So everybody, you do not shade with your pencil trying to shade, shade, shade. You join the lines. Spring will come on in the last part of class. He was here this morning as well. What is a composite function? That is where you take a function and put it inside of another function. And when you do that, you're making sure to simplify it. You can't just do it and leave it just so you have to simplify the relationship between a function and its inverse. How many of y'all have seen this before? I feel like it wasn't really like mentioned mentioned but you should know it that if you take a function and it's inverse you get x this symbol f inverse of x means the inverse and then f is the function right so the composite function f f inverse or you can do it the other way f inverse f that will give x no matter what the function is right that's just a relationship between it steps involved in finding the inverse of a function first step let y be equal to f ofx step interchange the variables and lastly make y the subject question. When I do the working, am I writing these things? A big resounding yes. If you don't write it, you don't get the marks for it. You have to write the statements when you are doing your solution. Right? I think you've heard that regularly. You have to even though in your homework I just be seeing things, people leaving it out. You have to you have to evaluate a function at a given value of x. This simply means substitute x inside the function. You write down the function. Substitute the value. Whenever you substitute, you use brackets and you work your way down to find the answer. All right. Just some tips. All the tips are in the notes. So last year the entire question K1 functions sometimes question four is split between coordinate geometry and functions. All right. So for the first part you have function f function h calculate the value f of four. Yes. You have to put things out into the universe. Zachary. So the whole question is going to be based on functions. The first step is to write the function f ofx is equal to 2 - 5x. Then we substitute x as 4. Everybody substitute it. Tell me what you get. 2 - 20 is going to give you8. Okay. an ice cream, right? H of zero. You if you bring functions, you are guaranteed to get this where you have to substitute. They're asking for h of zero. So, which formula are you going to use? Of course, you're going to use the h of x formula. So, you write down the formula. You substitute. What is anything raised to the power of zero? Knowing your laws of indices 5 to the power of zero is one. Nice one, right? Please put a one there. Fh of -2. So this question you have to use both formulas. Order is important. Are you going to do the F first or the H first? Which one is first? Very nice, Angelica. H is first everybody. You do the function that is closest to the value h. So h of x is equal to 5 to the^ of x. x is -2. So what is 5 to the -2? Using the laws of indices, when you put it in the denominator, you get 5 to the power of positive2 1 over 25. And right now, you're going to put it into the f function. The f function, I need to go back and see where it was. The f function is 2 - 5x. So you substitute, you work it out. You could put your answer as a fraction. You could put it as a decimal, right? But you know, we love fractions. So we'll use the fraction. So we have 2 minus. So I just waiting for the board to catch up with me. 2 - 5x. All right, just allowing some people to enter. Could we have it as the -2? Yeah, that's not a problem. Like if you put it into the calculator, that's not a problem. As long as your final answer is reduced, that's not a problem. So 2 - 5x uh 2 - 5x. We want to substitute 1 over 25 inside of this function. And everybody is giving me good things which is 9 over 5. 2 - 5 by 1 over 25. So 2 - 1/5 is 1 and 4ths which is the same thing as 9 over 5. So if you want to write it as a decimal it's 1.8. Yes. Good so far. So that whole thing was substituted. The next step is to find f inverse of x. So notice I have on the board already because I didn't I want to save time writing the tree statements. You have to write your tree statements. Yeah. Go ahead the h of negative2. So just now I just starting over the board cuz I just want it to be not stick. Not yet. Yeah. Uh so white bers get a little sticky after a while when you put much a thing inside of it cuz I would have used other applications but because I started over the So when I clean up the computer and everything, I take off all the plugs and stuff now. So it kind of like messed with with it. So this is the previous part. So everybody all everything is is in the notes, right? Don't worry about it. 2 - 5x. All the is telling me the answer for the other one. So everybody 2 - 5x find the inverse. Tell me what it is. Right. So hopefully you get the look just going across here. Now the first step everybody is to let y be equal to f ofx. So let y be equal to your function 2 - 5x. Then you interchange the variables. That means you're going to let the y be equal to x and instead of x you're going to put a y. Now you're going to make y the subject. So you bring a course. Well, to help avoid some negatives, you bring across the 5 y, it turns into positive 5 y. You bring across the x, it turns into 2 - x. And then you divide by five throughout 2 - x all over 5. And therefore f inverse of x is equal to 2 - x all over the five. [Music] question. Take a little walk along Liam. Sitting down being stagnant um will increase your fatigue. You have to walk and stretch. I think I have a question that somewhere you know. Oh, this one is f ofx. So actually this one is where you just have the composite function f ofx. So I'll do it I'll do it right after this one. Right? So I just want to finish up this question. We have the function 2 - 5x. We want to find the composite function ffx. So what does that mean everyone is that normally you're custom subseing values but you are going to actually instead put this whole expression inside of the function. So normally you would have put in like five like normally you put in a value but instead of x you're going to put in this whole thing. So it's going to become 2 - 5. Enter. You put instead of x you're put in the whole thing. You expand 5 + 25, right? Plus that's a 25. Let me fix it. 2 - 10 is how much? Thank you, Liam. It's8. You can check your back. Exactly. Yeah. Yeah. Oh, you said never mind. So, -8 + 25x. And do you all agree that it is in the form a + bx? So to answer the question, what is a what is b? You have to answer the question. That is something that's very important. you read back and you see did I answer the question. So a is8 and b is 25. So I just want to there was a possible question with it where you have to take the function and its inverse and prove that it will be equal to x. Right? So I want to do this one where you're going to substitute inside a function or that. So, I'm going to use the same thing. I have the function 2 - 5x and I just found the inverse 2 - x over 5. So, everybody, you have to pay attention to the algebra here. I'm going to take I am going to take inverse and put it inside the function. The inverse is 2 - x all over 5. So you see this whole thing in the brackets. Instead of instead of putting x, I'm going to put 2 - x over 5. Instead of putting the x, I'm going to put this instead. I'm going to put 2 - x over 5. All right. So, understand that this five is in the numerator and this five is in the denominator. So, that could cancel, right? Yeah, we are. It's in the measurement notes though. 2 minus you have to take very careful concern here. It's a negative. 2 - 2 - x. So when I distribute the negative, I get -2 plus x. And what's 2 - 2? 2 - 2 is 0. x is it's just simply x. So you will always get no matter what function you are doing f by f inverse is equal to x no matter what the function is. All right. So now we're going to go into some work with graphs. The all of these objectives is about quadratic graphs. This is composite functions. So that is why you're working in terms of x. If x is a number well you just substitute x inside here. All of this is about quadratic graphs. So when I say minimum, the name given to a quadratic graph is called a [Music] parabola. All right, let me just clear up something here. I distributed the one Kyle, right everybody? There's a negative one here. You had to do the distributive law. So it became -2 + x. That's why I have plus -2 + x. All right. Going on. If I tell the name given to a quadratic graph is a parabola. If I tell you to draw a minimum parabola, what does that look like? Like a smiley face. Yes, like a smiley face. Cuz you're being positive. The value of a everybody is a positive. Like a you positive. You're smiling. If I tell you to draw a maximum parabola that looks like a frowny face because the value of a is negative. So you're just being being negative. is negative. Right? For a minimum parabola, you will have a minimum turning point, right? TP for short. For a maximum parabola, you will have a maximum turning point which is the highest point in the graph. What is the formula? Like when you finish complete the square, what is the formula? putting fora in quotation marks cuz now you have formula. But how do you find turning points after you done complete the square? That's right. It will beh the y intercept. I don't need to complete the square to find this. When I have my quadratic, I already have the y intercept. We're going to we're going to do a question where you complete the square first and then you will see then you will see what I mean by hk. So when you get a quadratic, it's in the form ax^2 + bx + c. Correct? the value is going to give you the y intercept. Whether it's a quadratic, a linear, the y intercept is the value of c. But what if I ask you for the coordinates of the y intercept? What would you tell me? You have to read the question properly. If I ask for the coordinate, you would give me zero c. Yeah. Whatever C is, C could be 1, two, it could be anything. So X is zero. The Y intercept is C. Is everybody good with this? The Y intercept is C. Whether you have a linear or quadratic, this value C here, the constant Z or C. Yes. Nice. All right. We're coming up to a break after this section, everybody. coming up to our break. So I just have some ideas to understand first and then we do a whole question on it. All right. So I just have to draw on lines. I can't write just so in your question eight you can get different topics. You can get completing the square, you could get linear programming, you could get motion graphs. Now motion graphs the way the syllabus is set up I'm doing it in the measurement. All right I'm giving a 3 minute break and we'll continue after 3 minutes. Right everybody 3 minutes you'll get your 15 minutes after the section finishes. All right. Also, I'll see the comments from um YouTube. So, thank you, Cadia. Yes, the next break everybody. Um right now I have to do the maths to find out that break you know 245 345 I I can't do the math right now. My mind is is focused on on something else on making sure I have this set up. But once we finish this section, you guys will get a break. Right. We actually um rip through this section really quickly. I just have a couple more things to do with the quadratics. All right, everybody. I was on the call on my phone so it was echoing. So with this, let me just um a couple of the Google Drive everybody, the operations team is in charge of the Google Drive request, but you they work really fast. You will get access just request access. to get access really [Music] quickly. Yes, five more hours. Um, so like I was saying, this section, you'll completed it, you know, much faster. We have a couple more things to do with the quadratic graphs. Finish a question and then you will get your entire 15 minutes. So the first part when they when the first part that they test is for you to complete the square then they like to act different things. The different things could be what is the axis of symmetry. So that's this objective number 22. That's where we are at. What is the what is the axis of symmetry of your body? So it's an equation of the axis of symmetry. So you don't really want to tell them at just8. You want to give me an equation because if you don't give me an equation, it'll we'll have problems. So an equation have an equal sign. So that's right. You want to give me x is equal to h. It would be f of x. f of x is y. So you want to give me. All right everybody. So when you're giving me the equation of the axis of symmetry, you're giving me an equation. So you have to have a equal sign. Notice that if I were to split up the parabola with this line, don't I get two equal pieces, the axis of symmetry, symmetry, you divide it into two different pieces. Two equal two equal pieces. All right? So, the axis of symmetry x is equal to h. So, for all your your questions, you're not just telling me h, you're telling me x equals h. And then if they ask for the maximum or minimum value. No, this is not indicating the height. This is not an indication of the height. If you want this is this is the value of K which is the maximum maximum value of Y but X= H. Goldie that's simply a vertical line. The next part is you sketching. There are certain questions just last year that asked you to sketch a quadratic function. And your quadratic function to do so you need your roots. How do you find roots? We just did it in algebra. So the roots of an equation, that's where the graph cuts the x-axis. So you can factoriize and solve. Sometimes students use the complete the square form and equate to zero and solve. No problem at all. Whichever one you want to do factorizing, if you're really good at it, it's actually quicker. Place it to be equal zero and solve. At CC level, you will be dealing with quadratics mostly that have two x intercepts, two roots, right? At CC level, you you deal with two roots. And then you may be asked to sketch the graph of a quadratic function. So you want to make sure you walk with your pencil and Sorry, let me just snap it over. Uh, you put in the key points, then you draw the curve. For a minimum, you're drawing a smiley face. For a maximum, you're drawing a frowny face. And then you draw the x and y axis accordingly. So, if I have the roots -3, 1, you don't need to be exact. It's a sketch. But you do need to have negative -3 on the on this side the left and positive one on the right. Negative 1g4 is somewhere here. And your y intercept is -3. So those are the information for the quadratic x^2 + 2x - 3. So just to give you all a couple tips, when you're drawing the graph, you should use at least 80% of the page. Now, if you look at the questions, they tend to give you the graphs. Now, all right. So, you just give me a second. I'm snapping over everything and putting it into the board. So, it's right solving problems, right? Oh, that's when you we're getting to a question. I just did an I just showing you an example here. we get into our question where we're going to do it. So when you draw graphs you use at least 80% of the page you ensure to draw a smooth curve right not sketchy smooth curve and if you look at the recent past papers they are giving it to you they draw the x and y axis for you and in some past papers if they want an exact graph they mark in all the scales as well right so to determine the minimum point Elijah is correct ht Okay. So, when you take a look, we're going to take a look at last year, January. These questions, they show up in your question eight. Your entire question eight is based on relations, functions, graphs. So for a completing the square question, the first thing that they do is ask you to complete the square. You need to find the values of h and p. And yeah, you can use a calculator as well, but you still need to show your work. So you might as well do it here one time. What's the formula for h? B over 2 A. What's the formula for K? You have to know it by heart, right? K is equal to 4 A C minus B ^ 2 all over 4 A. So a popular question you guys were asking me before was do you have to use both formulas? Yes, you have to know you have to know you have to find out both H and K. So B is -4. A is pos4. So you work 24 / 8 it is -3. All right. Answers in the chat -3 is 31, b is4. All right. Thank you. four * 4. So you can put all of this in the chart in the in the calculator, sorry. What would you get? 4 * 4. Oh, thanks. calculator crew is making sure to give me good stuff there. H is -3, K is -5. So the last step is to write your answer in terms of the complete the square form. So you start off with the value of a. A is open bracket x + h + -3. Do I really want to do that? A plus and a -3 next to each other. I don't really want to write my answer like that. I'm going to combine it to minus 3 + k. So + -5 - 5. So this here everybody is the completely square form. Is everybody okay with that formula? Substitute. Work it out. Put it in the form. X minus, right, Liam? Yes. Great. So on the axis below, you see they drew it for you. A good sub. Everybody, you have to substitute. Work it out. It's that's just how the formula matter goes. All right. to take it in. Whatever is the sign of the h and k is the same for the equation. Answer the equation of what? The axis of symmetry. The axis of symmetry is x is equal to h. So this is h is -3. So h will be positive 3. So I'll show you right indicating the coordinates of the maximum or minimum point. First off, do I have a maximum or a minimum graph? Do I have a maximum or minimum? How do I know? I look at the value of A. A is positive. So because it's positive, I will be having a minimum graph. A minimum graph will have a minimum point and a is positive and I also need to indicate the y intercept. So looking at the quadratic, can anybody tell me what is the y intercept? The y intercept everybody is the value of c which is 31. Right? So we all agree it's a minimum and we all agree that the value of c is 31. Nice. We to know the coordinates of the minimum minimum point is in the form h. So we just found h to be3. So what is h? It's a positive tree. We just found k to be5. So I just put a5 there. So my minimum point is5. Chat, are y'all with me? Right. So I'm just waiting for my chat to load. H is positive 3. H is -3. So you'll have to make sure to be very careful of your signs. So the y intercept everybody is 31. Right? So I'm going to indicate those two things on my diagram. So my minimum point is 35. So you can plot it right here. It's a sketch so you don't have to be accurate. So three stop sharing. My screen stop sharing. Okay, y'all see me now. Okay. I could present over so that you could see it. So I'll know for sure that the um I'll just present it over. Let me just check on my YouTube people. Yep. Can you explain K as the constant? So what a constant means a number. So I just use the formula substitute find the value of K and I will get a number. K is -5. So I'm just going to replace it here. K to be -5. So my minimum point is 35. I I'm doing a sketch. So I plot it, right? Plot plot it um 35. And then let's say that this is 31. And I know I have to draw a minimum parabola. So, you know, you just want to be neat. And it doesn't that's the most neat y'all will get from me today. Y is equal um to 4x^2 - 24x + 31. Right? That is your minimum parabola. Right? I I'll have to take that. Make sure it's smooth. Make sure it cuts at the one and and 35. The last step is to find the equation of the axis of symmetry. To the equation of the axis of symmetry. So tell me what that is. You're giving me an equation, people. The theory is that x is equal to h. You just found h to be3. h is -3. So, x= 3. When you write your answer, very nice, Colleen, the um you write the equation x= 3 and equation have an equal sign and that is how the completing the square question can come. The first part is always to complete the square and asking you to do things sketch. So if it is you had a maximum parabola, it's the same thing. HK, that's your maximum point. HK, your turning point. So everybody, the time is 4:10. If I add 15 minutes to that, it'll be 4:25. You are to promptly return at 4:25 p.m. And we will go through measurement. I am walking away from the computer. So you're not you're not feeling to stay. We zoom in 425 break your resume at 4:25. We're going to do measurements. The SP Right, everyone. So, it's 4:24 p.m. Calling you all back to your seats. Hope you all had a that 15 minutes refreshed. You you walked outside, touched the grass, you come back and you're ready to Now now we're going to look at measurement, right? I'm going to take you all through measurement in the next hour and a half. So you have some entertaining conversations in the in the chat, right? All right, everyone. So, just making sure y'all could hear me. I have my camera mic camera mic. Nice. So, section four in your syllabus, it's section four based on measurements. However, it's tested in your question six. Whatever. Hear the stories about measurement man. How he just literally just love his job. make sure to bring the most do and do the most for the measurement question. He love his job. So the first thing we want to do is look at quantities and their units. So if um you look at length, length is length, distance, all of it is measured in meters. mass, grams, area squared, meters, volume, cubic meters, and capacity is measured in liters. So, Mrs. SM took you to some conversions is also here in the measurement notes. 1 cm= 10 mm. 1 m= 100 cm. You have to know this by heart. 1 kilometer is 1,000 m. And this is This is something that shows up in the the scale when you're doing scales. Um, 1 km is equals to 100,000 cm. For mass, 1 kg equals 1,000 g. Conversions based on time, 1 minute 60 seconds, 1 hour 60 minutes, and 1 day is equivalent to 24 hours. Conversions based on area. This is an area scale. 1 m² = 10,000 cm². 1 liter = 1,00 ml. 1 liter equals 1,000 cm. So you could understand that 1 milll is equal to 1 cm. Right? Those are some conversions you should know. You should know the SI SI unit is more of a physics term. What that means is just like a standard unit. So typically standard wise we measure area in squared meters, volume in cubic meters, capacity in liters, mass in kilogram, temperature in degrees Celsius, time is usually measured in seconds. Does anybody know the the SI unit for steam? Another word another Um speed is a scalar quantity. So the vector quantity is velocity and that is measured in me m/ second. Acceleration that's a vector quantity and that is measured in meters/ second squared. So if it is you want to write m slash s that's not a problem. um slash a squ nice so everybody need to find your keyboard and start typing in the chart these formulas it's not in your formula sheet so you have to know them so if I ask you perimeter of a square you add up all the sides s plus s plus s plus s so you get 4 s right four times s very nice everybody Ready? What is the perimeter of a rectangle? You're going to add up two lengths plus two widths. That's the same thing as 2 times length plus width. Some people use B for breath. That's not a problem. Area of a triangle. That is basically for you to add up all the sides. There are three sides to a triangle. A plus B plus C. For a trapezium, if you want to find the perimeter, again, perimeter means distance around the shape. So, you're going to add up all of the sides around the shape. For a circle, what the the specific word for circle, the perimeter of a circle is circumference of a circle. So, what is the circumference of a circle? There are two formulas. 2 pi r and there's one where you you um consider the diameter pi * the diameter. So the diameter is 2 * the radius. So I could replace 2 r with d. Right? So those are the two formulas. Pi is approximately equal to 22 over 7 or 3.14. When you are doing a measurement question, you are going to use the value of pi that they told you to use. All right? Uh there are certain questions in the past paper with compound shapes. And I just want to make sure you have the ideas behind this. If you want to calculate the perimeter of this shape, perimeter means distance around. When you're doing your calculations, are you going to add this 49 and this length? Are you going to do that? No, absolutely not. I've seen students be doing that. Don't be doing that. You want to perimeter. You do not include the length inside here. Perimeter means distance around the shape. So you're not including anything inside, just the distance around the shape. Do not fall for the trick where they give you the diameter and you did not remember to find the radius. Don't fall for that trick. Okay? Just look out for that. They most likely will give you diameter. You need to do the working. Find the radius and work with it. So you know the circumference of a circle, but what if I wanted a semicircle? What to do? If I wanted the circumference of a semicircle, I would have to do half of the circumference of a circle. Very nice. Right. Circumference of a circle. Forgot how to spell. No, I never corrected um papers. Let's see now. Right. So half of a um half of a circle half of the circumference of a circle. And what if the question had this? How would you find the perimeter? You would have to do half of a circumference of a circle. The perimeter I'm talking about perimeter plus the diameter. So you're going to find this. So a lot of you all know the word arc length. And this here is the diameter. So you make sure to really understand your measurement diagram to do your calculations properly. So what is an ark? An arc is a piece of the circumference of the circle. It's just a a piece, a part, a portion, right? A of the circumference. So you see what's highlighted in red? That's an arc. How do I find length of an arc? What's the formula? All right. So if you were given the value of theta, the length of an arc is theta over 360 * the circumference 2 pi r because it's piece or a fraction of the circumference and math staple. You know you're interfering with your calculator so you might put it in a different mode. Please put your calculator in degrees mode. Elijah, that's for adm. So yeah, you are to make sure to walk into the exam with your calculator in degrees mode, right? For us regular CC maths people, that does not concern us. This is for admats. In short to read your question properly, you'll need to make sure and look out for the words. Do they want the major arc? Do they want the minor arc? Right. Estimate the area of plain, shapes, polygons, circles. What's the area of a square? S * S, which is S 2. Lovely. What's the area of a rectangle? Length multiply by width. You could use breath. Length time width. Nice. There are a couple formulas for area of a triangle. If I have the base and the height, I will use a half base by height or a lot of people say base by height over two. No problem. Base by height over two. Yeah, same thing. But what if I didn't have a beta? I didn't have a height. I had a scaline. I had a scaline triangle and I had these two sides A and B and I had the angle at C. So, Angelica, we don't give no love to the heavens formula. It's in your formula sheet. We don't give no love to that. So, um what's the formula here everybody? It shows up. It can show up in your bearings. It may snagle some um something there. can a half a b sin c where a and b are your two sides and c is the angle here the included angle between a and b her formula don't worry you don't have to to um you won't be using that you most likely use one of these what is the area of a trapezium you all know cxc loves trapeziums not just in sea stack maths. But if you do like pure maths, they bring it everywhere. They just love trapeziums. It will be a half open brackets. This very nice Colleen, the sum of the parallel sides multiply by the perpendicular height h. You see this h here? That's the perpendicular height h. Not the slanted height, the perpendicular height. Great. What is the area of a circle? Well, from 21 reasons, not the parents. Anyways, area of a circle is pi r². Very, very nice. Lovely, lovely things. So people I hope you know your formula game is on that you're able to memorize and understand and apply the formulas calculate area of a sector. So sector pizza that's like a pizza slice. So the image in your mind should be like a pizza slice. So this here is a minor sector and the other part is bigger. So we call it a major sector. So hopefully you guys get the bigger piece right of the pizza major sector. So how do we find area of a sector? It's a fraction theta over 360 is a fraction of your area. Very nice to pi r². Ensure to read your question properly if they had major or minor sector. Right. Right. So let's um dip into our question. Area of a triangle given two sides and the angle formed between it. So what formula comes to mind? 12 a b sin c. That's the formula you're going to use for this question. A half a b sin c. Right everybody, you substitute a 10, b as 11, and your angle is 45°. Making sure that your calculator is in degree mode. Multiply by 10 by you plug it in and the universal way of writing something is to three significant figures. So very nice Ken. So K check it back right. Make sure you have this 38.9. And because we are in in in any topic you make sure to put your units it is squared cm to three significant figures. So tips the angle that you use this angle that we using in our formula it is the included angle 45°. Write your answer to two significant figures unless otherwise stated. Sometimes they ask for the nearest whole number. So it have been if they ask nearest whole number it would have been 39. Ensure to include appropriate units. So everything was in cm area will be squared cm. So why did I use sign? That's the formula ly. That is the formula that I'm using a half a sin c. That's via the formula. How do I find area of a segment? Right? Segment is the shaded part here. Minor segment, major segment. Segment is where you're bounded by a chord and an arc. So, the first thing you have to do is find the area of the sector, right? As a pizza slice. Then you'll have to find area of a triangle and then you would have to subtract it. So I'd love to see the correct things in the chart. You'd have to subtract it. Calculate the surface area of solids. There's a very interesting technique, right? Surface area everybody, you have to find the area of all the sides and add it up. Threedimensional objects. So for example, a cuboid height multiply by the width. You have two of those sides. I have the formula here. Height multiply by width. But you have two of those sides. Length multiply by width, but you have another side in the back here. So you have two of that. And then you have sorry length by height. And then you have length by width. And you also have two of those sides. So the formula is here. Per surface area of a cuboid for a cylinder. If you were to look at the net diagram, this is made up of two circles and a rectangle. Y'all ever had to, you know, carry this in school? You had to draw it out, etc., etc. Great. Right? So it's made up of two circles and a rectangle. This is the formula for a sphere. This is the surface area of a sphere. All right. So, Tatiana is just making sure you know your camera on. Type in the chat for me. Volume of a cube. If one of the length is S, what is volume of a cube? S to the power of three. Lovely. What is volume of a cuboid? Volume of a cuboid is length time width time height. And what is Elijah? What is volume of a cylinder? Right. So volume of a cylinder pi r 2h. So a volume of a sphere, volume of a cone, those are in the formula sheet, right? Volume of a sphere, volume of a cone. And if it is that I wanted to find volume of a prism. Well, a prism has a uniform cross-section. So it would be that you find cross-sectional area multiply by the length. Nice. So problems involving time, distance, speed, time is usually measured in seconds. Distance is measured in meters. Speed typically measured in meters per second and acceleration m/s squared. So you need to know your formulas. What is the formula for speed everybody? This is memorizing land. distance over time. So the definition of speed is the rate of change of distance. When you hear the word rate, that means with respect to time. So you're dividing by time. Rate of change of distance. It's distance moved per unit time. So distance is measured in meters. Time is measured in seconds. So you get speed measured meters per second. This is some conversions here. If you want to convert from kilometers per hour to me/s, you simply multiply by 1,000 over 3600. And you can just learn that technique off, right? I mean, kilometers to meters and then hours to seconds. Just learn off the fact that to convert from kilometers to meters/s, you're multiplying by this. To go the other way, m/s to kilm/ hour, right? You flip 36 over 100. 3600 over a,000. What is the formula for acceleration? So the definition of acceleration is the rate of change of speed, right? Rate means with respect to time. So acceleration is speed over time. Nice. So only C stack math formulas here. Zachary. Speed is measured in meters per second. Time is measured in seconds. So it's a little bit of what we call unit analysis. You don't need to know all of that. The unit of acceleration is meters/ second squared. So this these objectives you get tested on two types of graphs distance time and speed time graphs. You need to organize the information in your mind that you are not mixing up the graphs. All right. So if it is that I had a distance time graph and I found gradient what would I get? When I find gradient in a distance time graph I am finding speed. Lovely. If I see a horizontal line in a distance time graph what does that mean? Right? If I were to find yeah gradient no speed right so gradient will be zero speed will be zero so that means you're not moving right everybody you're not moving so what is the term I want you all to know stationary at rest the words the phrase that comes to mind you are at rest the object whoever is riding the bike you're at rest if you see a line that is sloping upwards That means the gradient is positive. So when you find speed, that's right. Speed is positive. But you're not concerned about speed is a scalar quantity. So it has no direction, right? Um speed is positive. Um when it is that you are sloping downwards, what does that say about the gradient? The gradient will be negative, right? So your your speed when you find the value of speed, the value of speed is negative. You're moving, you're just moving in the opposite direction, right? You're going down. You're probably going back home. Now keep that information separate and tell me now when you find gradient in a speed time graph, what are you finding? You are finding acceleration. When you find gradient in a speed time graph, acceleration here, what does a horizontal line mean? What is constant? The speed is constant. But what about acceleration? What happens to acceleration? Acceleration everybody is zero. You're not pressing the brakes. You're not pressing the X. You are going at a constant speed. You're cruising down the highway. You're cruising. you're not increas um speed is constant. When it is that your line is sloping upwards, what does that mean? That means you're increasing speed. You're pressing the the X. So that means the car is accelerating. When it is that you find negative gradient, right? You're slurping downwards. You press the brakes. So speed is decreasing. So that means you are slow you're you're decelerating right decelerating. Nice. So keep the information separate. Good. In a speed time graph the very interesting technique is that when you find area under the graph you are finding distance covered. Okay. And very typical CXE will test area of a trapezium. So please please please know the formula. I do not want to see anybody taking a trapezium splitting it up into two triangles and a rectangle and doing the most. Just learn your formula. So looking at a question that came in January 2025. Don't do that. January 2025. The velocity time graph. Now I know they use the word velocity but speed. Speed time graph below shows the journey of a car over a period of 70 seconds. You seeing that from the graph time is on the x-axis. 70 seconds. The journey is represented by five stages. One 2 3 4 5. Makes it easier for you to label. Complete the following statement. During stage four, the car is traveling at looking at stage four. The car is traveling at what? Well, look at the units. They want a speed. So, the car is traveling at you. Just read it off. Read off your scale. 40. Between 40 and 50. 45. Nice. 45 m/ second with an acceleration of you don't even need to do the working when you see a horizontal line the acceleration is zero. So knowing the properties you don't do any working you simply know that acceleration is zero. Part two, determine the maximum acceleration of the car during the 70 seconds. You have five stages. Are you going to do all the working for all five stages to find maximum acceleration? What's happening in stage two and stage four? The acceleration is zero, right? So, yeah. No, you're not at rest. You're not at rest. You are moving. You're just not you're cruising along the highway. So, that's why you need to everybody, you're not at rest. When you are at rest, that is a distance time graph. So, you need to keep that information separate. This is a velocity time graph. When you see a horizontal line, that means acceleration is zero. You're still moving. You're just cruising. You're not mashing the X or the brakes. Great. They want maximum acceleration. What is happening here? You are decelerating. You can look at the fact that the line is sloping downwards. So the value of acceleration will be negative. So you you are decelerating. So you're not even going to check that one either. You want acceleration. So you are checking between either stage one or stage three. Now let me see if some of y'all could even just look at it and tell me. Right. You need to look at the line for stage one and look at the line for stage three. Which one do you think will have a steeper gradient? If you could see it, are you all seeing that it's stage three street? Yeah. Yeah. You're seeing it right? Like when I say steep people, I mean um like if this was a hill and you had to walk up this hill, which which hill would be pressure to walk up? This one will be more pressure to walk up that hill, right? It's more steep. So really your maximum acceleration will be in stage three and you need to determine that. How would you do that? You're not looking at the length, right, Janelle? So everybody that's that's important. You're not looking at the length. You are looking at the line and how steep it is. So this line is have a gentler slope. This one have like a steeper slope. Very nice to find the gradient. So to find the gradient, you're going to pick two points. Please don't complicate your life. Just pick the end points. Don't pick anything between here. read off the x and the y. So 25 20 x is 35 45. Right? Right. So to find gradient you pick two points 25 20 35 45 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3 45 45 Great. And now you find the gradient. So you're going to use your very well-known formula y2 - y1 over x2 - x1. Please substitute accordingly. y 2 - y1 x2 - x1. So you're going to get 25 / 10. Do we leave answers looking like that? We make sure to reduce 2 and a half 5 over two. The decimal is [Music] 2.5. So um Nisha the midpoint is the unique formula with a comma. That's the only formula you will know with a comma. The gradient you're using it all the time. Change in y over change in x. So 5 over 2 decimal is 2.5. What is the units to go with this? I am finding acceleration. So looking at the units here me seconds me per second squared right that is the acceleration sorry so I would a minus before the oh over here so if I remove the slash I have to put a the power to the -2 but if it is I put a slash um slash square. Yeah, you could do that as well. Where do you get those points from? These points are taken from the graph itself. So I chose the end points, right? You're looking at stage three. So I chose the end points and you read enough X Y 25 X Y 35 45. Uh choosing random points. No, I don't want you choosing random points. I want y'all choosing the end points. You cannot read that off accurately. So I want you choosing the end points. You'll be accurate. 2.5 accurately. Yeah. Go ahead. 25. So between between here 25, you're reading off your scales. Between 25 and between 20 and 30 is 25. All right, everyone. So that is that part. Last part is to find the distance traveled during the first 25 seconds of the journey. How do you find distance traveled in a speed time graph? You are going to be finding area under the graph. Right? It's a very unique concept. Area under the graph, but you only want it for the first 25 seconds, right? So 0 to 25. So what does this form? That is your trapezium. So you're not going to split it up. You're going to find everybody area for trapezium. The formula that you are learning off is 12 multiple sides multiply by the perpendicular height. These are your parallel sides. So between 0 to 25, this here is 25. Between 15 and 25, what's this length here? This is between 15 and 25. So that is 10. 5 10. And what is the perpendicular height? 0 to 20. Lovely. So, make sure to substitute appropriately. 12 10 + 25 * 20. And what are you guys getting? A half by 10. 10. 315. 315. Now I know we're finding area but really this is distance traveled. So what is distance measured in here? Distance traveled will be meters. Right? So I know we found area but what you really want to write is meters 350 m cuz that's what you found. Distance traveled it it would be me squared. Distance is a scalar quantity. So distance is meters. Just meters. I know we found area but it wouldn't be anything with area. It'll be distance equals 350 m. So that there is and doesn't matter which 25. No, it doesn't matter because addition is commutative. So it could be 25 + 10 it will still be correct. So that's a motion glass question. Um, margin of error doesn't show up in your CEC maths, but it's in your syllabus. So, let's take a look at it. What this is is a range of values within which the true value of the measurement is likely to fall in. So when you do your questions, right, CXE sometimes gives a little interval. Correct. All right, let me just go take a little quick quick thing here. Everybody reading off the points. When you read off points, you read off X first and then Y. So just to make it a little different, if I asked you for this point and this point, what would it be there? Faith, everybody, if I ask you to read it off, you're reading off the X. So your eyes have to track it. I know I know you're watching a screen, but your eyes have to track it. This is 50 and then this is 45. So the point everybody is 50 45. So I see Elijah has it. Okay. Nice. Exactly. This point you read off the x as 70, the y is zero. And then you could use those two points. Okay. So, let me hold on the the recording. So, I have the controls to to put it back on. Just give me a second. Let me just cancel this. Cancel that. Great. So, big up yourself. It has been 8 hours. The recording times out at 8 hours. So, y'all have um secured 8 hours of the crash course. Y'all have been here since 9:00. I seen the names. Some Y have been here since 9:00. Y'all going strong. I just wait for the recording to start before I continue. So, so yes, 8 hours to put a check box and you did sections 1, two, three, you did section nine, you did section uh six, seven, and then this is section four. So, you're making sure to to go at a steady pace. Yeah, make sure you eat and make sure you sleep. It's um you do need to sleep in order to retain your memory a 9 to5. So just making sure everybody is ready. So, it have to run through the whole desk and um then it'll start to record. So, if it get booted out, join back in. All right. It might boot out some names, but join back in. So we'll definitely finish up a little earlier with the measurement. Um so Mr. Springer will get more time to do the like the geometry and your transformations and things like that. Right? So we we're going forward with it. So if I asked you to find deceleration, you would use the two points, find the gradient, you will get a negative value because it's a line sloping downwards. All right? So make sure to read off nice. Make sure to read off your points accurately. So as I was saying, margin of error isn't tested like I have not seen it tested in the past papers. But we're just going through it because it's in your syllabus that if I say the measurement is 30 cm, right? And you have a margin of error of 1 cm. that range of values that um range of values would be between the 30 plus the one and 30 minus the one cuz you have a margin of error of 1. So it be 30 + the 1 30 minus the one. So your range your interval will be 29 less than or equal to x less than or equal to 31. Right? So that is what margin of error means. It's a statistical term. [Music] Miss you could explain this. So everybody, if it is that I said that the if it is that I said that the measurement is 20. All right. If I said that the measurement is 20 and I'm telling you that the margin everybody lock in, don't worry about it. If I tell you that the margin of error is five, right? So everybody focus on the board. Now the measurement 20 cm and I'm telling you that the margin of error is five. that interval your interval would be between 20 + 5 and 20 - 5. So when it is that you um work it out you get 15 less than or equal to x less than or equal to 25. So your true value will fall within that interval. Nice. So moving forward because this is not reallyed thing. Let's look at scales. So the scale is 1 to 50,000. This means that 1 cm on the map is equal to 50,000 cm on the ground. Everybody locked in. Quiet the chat. Everybody quiet the chat. 1 cm is equal to 50,000 C. So this is on the map. You need to make sure that you understand 1 cm is on the map. 50 cm is on the actual ground. All right? So they use a scale so that you know geographic people will be able to understand this um way more. Admit admit Right. So, we have two towns on this map. Town P, town Q. Yeah, this is my little map here, DL. That's that's um the island there, right? just right. So, I just making sure that Google Meet could see my board. So, that's the that's the island there. And those are the two towns. Town Q. Just just take it in. If it is I ask you to find the actual distance from P to Q, we will only be looking at straight lines, right? The actual distance from P to Q. How many cm? 1 2 3 4. That's 4 cm on the map though, right? 4 cm on the map. So how much will it be on the actual ground? We already found for one. To find for what you want, you multiply. What's 4 by 50,000? That works out to be 200,000 cm on the actual ground. Yeah. Go ahead, Rachel. The um comes before simply for full squares or even if it's not full squares. Oh, actually this is just distance distance. So like this is one, two, three, four, right? Four um cm across. So that's how I got the four. It's a full square. It's a full 1 cm. Two, three, four. It's a full length. Four 4 cm of course. And all right. So that's the actual distance 200,000 cm. And just as a um quick question, if I want to convert this to kilometers, what would you tell me? And how would you how did you get that? Um divide by 100,000. Nice. So the conversion is that 100,000 cm equals 1 kilometer. So some students know when to divide and when to multiply. But if you don't know when, this is what you do. You rely on the keys that you learn to ratios. Find for one. Finding for one is a division process. Right? Well, some students know that. Some students know and and understand when to divide and multiply. But if you don't, that's why we teach the keys in ratios. So the keys are find for one, finding for one, you divide, finding for what you want, you multiply. So you'll take 200,000 multiply by 1 over 100,000 and you will still get the same two kilometers. This is part A. Go ahead, Pablo. Repeat. I didn't I didn't hear what he said. All right. All right, everybody. Um to find actual distance. All right. All right. Take a deep breath, everybody. Let's just finish this. All right, everybody. Mute your mics. Mute your mics. You're going to use your chat to talk. All right, people. Please don't understood my class for the remainder of the time that I'm here. Thank you. All right. So, everybody, um, you're going to for me to go and find the name and and remove. I will take a five minutes and do that. Right. So I just want to while I'm doing that I want everybody to take a look at the question and realize that I'm doing part A. For part A I have converted cm to kilometers. That requires the use of your conversion. Your conversion rate that you are learning off is that 100,000 cm is equal to 1 kilometer. I want everybody to now have a quiet moment and take the opportunity to do part B and type your answer in the chat. All right. So, I'll go straight after everybody do part B. All right. Excuse me. Excuse me. Everyone, please um refrain from putting on your mics. No one is on their mics for the remainder until 6 p.m. Understand? I know. Just give me a second. I'm tining the names. It's a It's a list I have here. I know what is still on their mics. You have to use the good students. You have to be doing the question right now. I just got an admin on board to to So if it is that I ask you to find the area and cm squ what are you doing? You are to be finding the number of squares that you have in here. And I know that that is something that reminds you of your form one days. So and yes that comes in C stack mats. Please upon your mics at all everyone a you could type in the chat. So when it is that you count up the squares in there. The first thing that you do is that you count up the whole squares. So the whole squares you have one, two, three, four, five, six, seven, eight. You have eight whole squares. Everybody should be paying attention on the board. Hate eight whole squares. So when So when it says that you find the whole squares, now it's time to find the piece of squares that make up a whole one. So you can say that this one and excuse me um Muel sorry but you'll have to type in the chat. Sorry I know but you'll have to type in the chat. So um a half of a square a half of a square and makes up a little nine. Then this could make up a little 10, a little 11, a little, you know, take that and make 12. So everybody getting 12 drop a chart. If you're getting 12 cm squared, you're finding area. So you have to find the area of the number of squares. So that will be 12 cm squ. So to what is represent what is the area in cm squar represented on the map? The answer everybody is 12 cm squared. Right. The answer to part B is 12 cm squared. And that's the answer to part B. So just talking back through part A, the first thing that you have to do is find the distance. The distance is 4 cm across. Why is in the chart if you understand that? Nice. So 4 cm is on the map. You need to take the scale and find it with respect to what is on the actual ground. So you're going to multiply four by 50. All right. So So my mic was off. So I don't know when that happened. So just to repeat, I found 4 cm. Correct. Now I multiply by 50,000. I'm going to get 200,000 cm that's on the ground. You divide by 100,000. So you're going to get two kilometers. Great. So what we want to do now everybody is this is the last objective. locking everybody for the area on the map. Correct. You're counting the squares. So the area on the map is to count the squares. And the other one where you wanted to know the distance, you're just counting each space as one cime, right? Exactly. You get it there. Yes. That is why I was trying to differentiate. Okay. Yeah. Distance just across and then area. You count the squares. So this is the last objective of the measurement and then we will wrap it up with our question. So the tips that you all need to remember when you're solving problems involving measurements, right? So my mic is on, my camera good. Everything should be good there. Liam, the first thing in any measurement question, right? This is what you I know the measurement man be doing his thing, be doing the most, but this is what you're always going to remember. formula, substitute, work it out. So, whatever formula you look at the question, whatever formula pops in your mind to apply to the question, you write it down, substitute, work it out, and in your final answer, you make sure to write your units, whether it's area, volume. If you don't have units, you put the word units, right? Um, I'll go back to part B just now. In certain questions, if it involves the use of pi, you're going to use pi that is I'll explain part B after this, everybody. For the value of pi, you're going to use the value of pi that they gave you. Whether it's 22 over 7, 3.14, you're going to use the value of pi that they gave you. You're going to read your question properly to know did they ask me to put it in a particular way two decimal places three significant figures they they typically ask to the nearest whole number. So you're going to make sure to read the question to do what they asked of you. Work with the this is a very important point here everybody. Work with the entire length of the number throughout the question. The only time that you approximate is when you reach your final answer. If you approximate too early in the question, that will throw off your final answer. All right? So, make sure that you are using the full thing. Even though it's like probably really big, you're using the full thing until you reach the last final answer. So when you take a look at the recent past papers, you'll see measurement manders be putting out things like hemispherical bowls, trapezoidal prisms, that's last year question, spheres, cones, semicircles, and then all the basic regular shapes. So you need to really have a good understanding of it. Hemispherical, it was a bowl. So it was for you to find volume of a a sphere and then divided by two something like that. Ensure to have a complete understanding of the use of the formulas, right? Practice questions with uniform cross-section as well as ideas behind capacity and density. So, so that is what is happening in the measurement question. I want to do the question that came this year January 2025. See how they said use pi to be 22 over 7. So, we have a piece of wire, right? Time to grasp the the information from the question. We have a piece of wire that is 61 cm long. You're going to bend it to form a sector. So, sector pizza. I agree Josiah. As shown in the diagram below, the sector of the circle O P or O P Q has the center O and the radius 18 cm. So very nice that if this side is 18, this side is also 18 because they're both radius. Show that the value of X is approximately 80°. So this is 18. This is the 18. How can I find this? I was given the perimeter. So, y'all putting good things in the chat. I see nice things that I want to see in the chat. So, the length of the ark, I will take the perimeter 61 and minus the two 18s. So, what is 61us 36? Who helping me? Thank you, Elijah. I was just double checking. You're correct. 25. Everything is in centimeters. What is the formula you know for length of an arc? Your cack maths formula is theta over 360. Right? So like circumference 2 pi r. So that's what you're going to use. You're going to equate this to your 25. So theta over 360 * 2 pi r. You're going to equate it to your 25. Substitute what you know. You don't know theta. You're trying to find it. Pi. What are you using as pi? So Tiana the 61 came from the question 227 is the value of pi. So pi is 22 over 7. The radius is 18 and you're working your way forward. You could put all this in your calculator. How much is that? H 18 right 7 92 over 7. You are not approximating early in the question. You are using this entire thing using the exact value. So I really want you all to put the fraction seven. So you put in the whole thing. Right? Now I need to make theta the subject. When I bring things across the equal sign, the operation changes. So it change from multiplication to division. Your calculator will handle that. 25ide by the answer you just found out to be. So um that's wrong 25 / is equal to that leave it alone 175 over 792 right theta over 360 is equal to 175 over 792 when you bring across that 360 it turns from division to multiplication multiply by 360. So this is your answer. Multiply by 360 and you are approximating. So approximately if I were to approximate this to the nearest whole number. It's a show that question I want to show it. So approximately 80° 80° to the nearest whole number. And when you show something, everybody is very very satisfying to write QED. I cannot tell you how satisfying it is to write QED. It is you did a whole bunch of work and now you have to write QED. Um, my students will know about writing QED. Excuse me. Excuse me. I no exc um somebody could mute it for me because on my side it's a bit sticky and um um I don't want anybody turning on their chat until um Mrs. Pinger comes back. I would like to do this. I would excuse I I would like everybody to type in the chat. Sorry that it inconveniences you to not p your mic. Um I'm almost finished. So if it is that we could have the decorum until I am finished. So this is part one everybody. This is a very integrated question. So you are Oh sorry Mr. Springer I thought it was somebody else. I cannot see who's talking. Go ahead. Sorry. Sorry. Um sorry about that. So either There was some people who were making out. Yeah. You know, it happens around this time. Yeah. Um there's not much I could say to that if you obscene um and you lose access to the crash force and everything else that's on you, right? um the admins will have to remove you and um the other thing I could say to that those of you going to class don't let the behavior distract you and don't let it also don't let it also feed into their attention because the number one thing some stuff went down a few minutes ago somebody texted me so the number one thing that someone like that needs is the attention. So we just um going through now to make sure all the right emails or the right students are in the class. There may be a couple students who may be under the guise of anity probably didn't get a chance to express themselves any stage. This is the stage where they think they can express yourself. So this is the stage expressel. So um I'll just turn on my mic on YouTube. I'm just taking a two more minutes and just saying um students please um if you know that um some of your friends are doing it last year there was one person um I won't call the name of the school to bring the school in dist um I I don't have much to say about that behavior. You just have to act accordingly. It happens in big lessons classes as well. So like even if I have a big physical crash course before there's sometimes somebody who's you know deprived of attention and feel that this is the best time to give trouble. So those of you all who have been here a long time please don't give this person's attention. I'll be hanging around um as we just sift through to see if there's anybody who's who has to be removed. If you have been making obscene comments and you are removed, that's on you. So, right, we can't have that kind of behavior. Um, yeah, you are only embarrassing yourself. Okay, love and blessings everybody. Now, if there's anybody who's trying to get through now, because of that, we have to make the admittance a little more strict. So, if there's anybody who lost contact and is trying to get through, um, M, you could you could share back your screen again. If if there's anybody who lost contact and is trying to get through um you could let them message the admin directly so we could accept them in the group. All right. So I'll be hanging around watching the chat. Love and blessings everybody. Enjoy the rest and that. So everybody should be hearing me and seeing my screen. Just trying to draw a little diagram here for you guys quick quick. Um right and I just want to represent on Google Meets. So we just talking it out that we found the angle 80°. Everybody good? Found the angle 80°. Some computation some algebra skills there. Just I just wait Google me to pop back up on my side. Mini, I'm not seeing you on the Google Meet. Yeah, the Google Meet is um Let me just give her one minute to see otherwise I just jump in one time. What What topic are you on? It's just to finish the question. You can finish measurement. The measurement question. I'm disappointed about some of the students who are reacting like that because come on, two days before the exam Let's see if I can get this screen. I'm back in um presenting the screen. So, you should see my screen pop [Music] up. It should pop up there. I presented it. It should be popping up now. I am not seeing your presentation. I don't know if anyone has seen it at the end. Yes, it's coming up slowly but surely. Okay. All right. Let me come up. Maybe it's that I'm here that it's all right. All right, everyone. So, you should see the board now. Everybody should see it now. where we need to find area enclosed by the wire. So that means you need to bring forward the formula for area sector. We found the angle to be 80°. So now you're going to write down the formula theta / 360 *<unk> r 2 theta is 80° over 360. Pi is given as 22 over 7 and your radius is 18. So you plug all of that into your calculator. Who's going to tell me the answer? Multiply by 22. multiply by 18 squared. So you're going to represent your answer. They didn't say you could represent it to three significant figures. Um 2 26 to three significant figures. All right. So you're representing it to two decimal places. The universal ways to three significant figures. But you can put to two decimal places 226.29 29 because we found area the units will be cm squared. So you just put in brackets what you did to two decimal places. Very nice. Let's take a look at part B. We have a cylindrical block of cheese, right? and it has a radius of 12 cm and a height of 8 cm. The cheese is divided into equal slices. The uniform crosssection of the cheese is a sector. Right? This is your sector whose angle is 18° as shown in the diagram. Everybody interpret the diagram properly there. Sector So calculate the length of the ark AB. This part here, the ark AB. So you want to um recall your formula length of arc. So just chuck it back, Goldie the over 360* 2 pi r. So theta is 18° over 360. Pi is 22 over 7. And your radius is the 12, not the 8, 12. So when you plug that into your calculator, make sure you're using brackets if you have the sharp * 2 * 22 / 7 * 12. So we have to three significant figures everybody. That's the standard way. 3.77. What's my units? Everything here was in centimeters. So length is in centimeters to three significant figures. Formula substitute final answer width unit. That is what the flow of a question entails. Determine the area of the curved face A C D. Now I know it's curved but this is a rectangle that you're finding. Area of a rectangle. All right. Area of curved face. Area of a rectangle, right? You put the rectangle in quotation marks, but it's it's area of a rectangle that you're finding. So for area of a rectangle, you need length by width. This here is eight. Sorry. And the this here is we just found it to be 3.77. So you're pressing the SD button for decimals. So area of a rectangle length times width. Length is the 3.77. The width is 8. So plug it into your calculator. um 3.77 multiply by 8 and you could represent your answer as exactly this 30.16 equate to three significant figures. So 30.16 we found area so what is what will be my units cm squared right 50.16 cm squared and we have a last part just need to go find it the last part of the question is to find the volume of the entire block of cheese right so they give you some information you're given that area of O A is 22.6. So I just going to go back to the diagram so you know what O A is O A is the area of the sector. Yeah. O A is the area of the sector. So they they gave you that area of the sector 22.6 cm squared. We were also told that this is a uniform cross-section. O A B that's a cross-section. So if I want to find volume of the slice of cheese, what do I do? This here was given as 22.6. So to find the volume of the slice of cheese, very nice cross-section by the length. Exactly. Crossctional area multiply by Well, sometimes it's length. In this case, it's height. You could adjust the formula to to what you want. So the cross-sectional area is 22.6. The height is 8. Plug it into your calculator using the entire thing. 180.8. Very nice. What will be my units? Because we found volume. It will be cubic, right? CMQ cubic cm. We're not finished. Find the volume of the entire block of cheese. This is just for one slice. One slice is is for your 18°. Right. 18° is for one slice of this cheese. How many slices are in this block of cheese? How are we going to find that? Everybody. Well, how much is everybody? Right? Because 18° is for one block of cheats. How many degrees does the entire block? It's a cylindrical block. How many degrees for the entire thing? 360°. So just by the way, I use height because my cross-sectional area is the factor multip and so the the volume is cross-sectional area times height. So the entire thing is 360. One slice is 18°. So I have to divide. So just switch it around. Celeste. 360ide by 18. And I am going to get 20 slices of cheese. 20 slices. So the volume of one slice is 180.8. The volume um of 20 slices will be how much? That's Yeah. 20 cheeks. So to find for what you want you multiply 20 ultip by 180.8 how much is that calculator is telling me 3616 niceness and that will be cubic cm. So that is the idea behind cross-section cross-sectional area um that idea length of an arc area of a sector. So that is what the entire measurement question was based on um in January there and scales margin of error. So that there concludes the section of measurements. Everybody feeling confident to do their measurement question now? period of time. But now I must bid you a due hand you over to Mr. Springer to take you through he's going to take you through statistics geometry trigonometry. Yeah. Um miss thank you so much is one of our qualified teachers at the student. She teaches the pure maths class. She has her own CC maths class as well. A lot of you all some of you all may actually be taught by miss um yes um she also some of the notes that you see generated like some of I I believe there's notes this would have been transcribed by miss two years ago on am I correct from one of the crash courses I did. So y'all could thank miss for the well-written the well typed up notes. A lot of the stuff that you see around me student um miss has done some of the stuff behind in the background. So um even in class some of the class notes that you have seen would have either been tweaked by miss or typed up by missed edited by Miss based on what we do in class. So a lot of your um success is due to a lot of the work in the student hub by different persons and Mr. Nolini is one of those persons. So thank you Miss Nolini. I am very sorry that you had to deal with some um reckless wire horns a couple minutes ago. So hopefully I think we we're out of that and thank you for the rest of the students who were behaving really well and you know um give Miss Undivided attention. Thanks miss. So now I am I I actually took a partner. So I am coming in fresh and I able to handle the rest of this. But let's take a vote. Do we just jump right into it or do we I'm just removing your screen and the me and removing you from this. Right. So do we just jump right into it or do we take a little break at it? I can't comment to the break. So on the schedule is actually a 15-minute break. So we break until 6 until 6. See you at 6 p.m. However, I'll be here. I'll be here. I'll be here taking any comments. Just setting up my stuff, making sure everything is everything. You know, we have to jump into statistics now. Statistics is on page something. There it is stats. So you have to jump into statistics. And then what do we have? We have geometry and trigonometry. That'll bring us to the end. So statistics is a really stable topic. It comes all the time. All all at the time. Yes. So you have a break until six. So grow. I don't know. Maybe you was glued to the screen and you still have to eat or take a little bath or something. Maybe you beat for the day. I ain't going to take no votes on that. So great. They smelling construction until 6 p.m. did Elijah G. You said like I never use a compass before. [Laughter] Fascinating YouTube people and those of you who watching over the recording because a lot of you will be using this link to watch over the recording. I hope you all are doing well. I hope you all are getting prepped up for the exam. Is there are there any major exams tomorrow? is gone TD. So yes, something to use your compass will very likely come. I know I have been telling you all that construction has been on a hiatus, but that doesn't mean using your compass has been on a hiatus. That doesn't mean we stop using compass alone. Where what what has replaced construction folks? What has taken over that that construction transformation geometry? So it's still there. It's still there. series and sequences that was missed. Oh no, I touched that. You see that's how series and sequences comes in the syllabus. Liam, so what this crash course is a topical crash course, right? So we have 10 more minutes to the break. Um this crash course we go through topic by topic right class. So y'all people think what they do on the internet anonymous. We have on a feature here that records all the emails and it's the same thing people do last year and it disappointed me to see some of the names that I mentioned um don't give them that kind of um attention. So thanks Yeah, I heard somebody else playing music. All right, let me get back to the um thing. So I I don't mind a little construction. I hope she sequences come. It does always come. So yeah, something you're using your compass will always come. It is I see a bunch of people were asking about the calculators that many of you all messaged me privately. Actually, it was so strange. A couple people messaged me the same time, sir, I need a calculator. It reminds me of my days in school when people were asking um were asking before the exam say I walk without a calculator. So I was wasn't I saying all year to get the calculator since since July last year you can't get this calculator from the student up in time. I wouldn't want to sell you it at this point in time. you wouldn't get it in time to get the student of if you're interested in that Karen Jeremy if you're interested in the student of you just WhatsApp the number and and message them and they'll give you the early so the maximum calc to walk in the exam I think they might be suspicious if you're walking with more than three I think they watches strange with three actually everybody a lot of people use two so two is not a problem why are you going with more than So don't believe the rumors, your calculator doesn't just give out just soon. Once your calculator has this solar panel on it, it doesn't just especially if it's a Casio. It's not like you're walking the exam and it stop working. These calculator don't ever happen ever. In all my years of teaching, never seen that happen with a calculator with a solar panel. Those calculators dim. They get dimmer and dimmer and dimmer before the battery dies. So if your calculator right now you had to press on a few times and hold it a few times in a special way to do the calculation. Well yeah I would go in the exam I would go in the exam with that and I've seen students having to I myself had a calculator like that that I went in an exam with and it still work to tell you I just need I know I had to hold it a special way now but I would advise that you do that. Um so we have how much? 7 more minutes until 6 p.m. where I'll start off. I don't want to talk anything important. Some people may be actually using the break. Some people may be taking a quick bait to freshen up. Sometime a bath hits the spotter and I a little cold water and all of a sudden your brain start back working. Some of youall might have to wash off the the um the messages from those people who needed to touch grass. You know, shake off some of that. Get back to equilibrium. only have seven more minutes though if you're going to take a bath. Some people that might be, you know, if you guys make sure to hydrate. Yes. Drink some water. Is what? What was the question? No, no. I've known people use pencil and get one. But just make sure your pencil has smudging and it's it comes to you clear. Tig running up the road for Tig and hydrated with monster. Yo, I don't really I know some people do it, but I don't really advise using those kind of drinks like Monster and Red Bull and it's sometime apple will do the trick better than a monster. Try it. Those of you who like monster craving for the monster water, coconut water, vitamins, a little porn apps here and there. That does the trick. Healthy eating sun, a little bit of sun. Right now, sometimes when people studying their body, they don't get no sun. Take a little walk around outside, smell the roses. Sometimes somebody some people act they're not like this. They're not usually like this. They normally good well behaved citizens. They don't go on the internet and bring down the reputation of their whole street and lineage and family and generation. But they gain enough sun. They're just inside all the time. They even studying properly. So they decide to, you know, come and get trouble and life. So yes, I have a 12 hour for adm um so doesn't have the thing to put in fractions. What chandler? What type of casio? You know how strange of a casio you have to buy to not have fractions. Even the casios that are not so hot have fractions. Reno from Guyana, where's you seen my brother? Pen or pencil? I just answered that question. Excuse me. I advise working at a nice pen and working accurately as possible. But some people work with pencil all the way through. And as every year students ask me that same question. Those students work with pencil in the exam. Boom. Shot. Them students get one two. Maybe there are some students here who did January exam and can testify to make you feel a little more at ease. They would have done it with the um with the pencil and they get you. Emma, what's up? Elijah, what going on? Where's the love? Where's the love? All your little auntie. Mel, I see you was turning on your mic. Mel, what going on with you, Mr. Mikuel? AJ, what is up with you? All right, be yourself out in these streets, right? So, my casu just arrived. You have any YouTube video tips and search curio online. Don't let them boys there. Don't let them boys there. Rescue them coh problems. But and we're going to start back in the next three minutes. The physics crash course is topic based and I will try to stream it on YouTube as well. I'll be working that along with Miss [Music] Christique. I don't know anybody could get the Casio this last minute. Um just use the calculator that you're accustomed with at this stage. Uh so we don't do one we don't do immediate deliveries in the student hub. It's too dangerous in this climate. So what we do is um delivery through post and that usually takes a week. So that's for people who was ordering calculator big up yourself if you order the calculator last year. I just put this here because in an online class you see all kind of things you know what's the chances of seeing this boy so I need to check your back my fish I need to save my fish life just see my fish life what's the chances of that is really encounter the best of the best I hope she get you with them fish All right people it's time to lock back in two more minutes two more YouTube people, you all should be getting ready. Big up, big up the YouTube family. YouTube family just quiet right now. I streaming. They're taking notes. They're going and studying. Make the most out of this. So, it's important to keep a very colorful mind when you're studying still. So studying if you treat studying like oh this big serious get thing and tensely writing down everything you going to remember much it I I believe when you're studying it should have a degree of color and a degree of humor even even in teaching it should have it should it should mean something so each topic is it goes down a little easier a 12-hour crash course will be very tough a 12-hour crash course will be very tough without some you know sometime even the little incidents too. You remember it in the exam? Oh, yeah. And when when M was saying that so and so say this. Okay. Um All right. It's time to log back in. So, we're going to pick back up with statistics here. So, we're going to have a little crash course on statistics and then geometry and trigonometry. Uh let's go differentiate between sample and population attributes. So, I'm kind of browsing past this. Let me just make sure my screen is up on YouTube. YouTube family say yeah. Yay. All right. population is the entire group. Sample is where you draw from the population. There's more applicable for the SBA. Of course, if we want to determine the attractiveness of men in Trinidad and Tobago, we wouldn't just watch one person and say, "Whoa." Like, you wouldn't just watch me and be like, "Wow, Trinidad had the best looking men ever." You got to take a You had to take an average of the entire population, right? You had to go and check everybody. You had to watch the M, the Simeons, the Gibs, the Jack, whoever. And you build your average out of that. And from that, you can say your inference and say, "Well, it's true. It's true." Well, not as good as sir, but they get in there. Get in there. So, that's what that is about. Construct a frequency table. You all know one of the main things that pop up in the frequency table is this group data stuff. So, you all know where it's going to end up. Eventually, we're going to end up f(x). Then what? Everybody know how to do this now? FX sum FX. What it is you're doing when you see this frequency table? Why is in the chat if you know about this? N if you don't know about this. Let's see what going on here. Why for yes if you know about this and if you don't. My camera start falling back. Okay. Just now. Yeah. You need a little clarification. Okay. That's all I'm just freeze back my camera. Okay. So, this we're coming up to this just now. You all remember this. This is very important. it shows up in the exile. We are working on the mean. We are working on the mean. For those of yall who see my hand go up like this, I know I'm not saying the pledge. I'm just trying to get my camera to behave. That's the way to do it. I tell it to stop. Um, so let's go through all this definition and stuff. Continuous variables on group data. You all could read up on that. Nothing too big straight up there. When CXC CXC say group data, they're not really talking about group of reds and blue. They're more talking about this class. You all have an understanding of this. Recently, they have been whining with this. They have been giving some really weird um intervals like they've been doing like 11 to 15, 15 to 19, 19th or whatever. You'll you'll see some of these intervals. We saw some in the crash course. They've been messing up this whole interval theory. In other words, they're using the boundaries instead of the limits. But usually, no is no limits, Elijah. They're using just the boundaries. Usually it's like this 11 to 15 and there's a break in between. So we call these things the limits and the boundary the upper class boundary for the first class interval will be what here class talk to me 15.5. You all understand? And then this will be like 10.5. 15.5 will be the boundary here. and then 20.5 will be the boundary here. So it is possible that instead of using the limits they just use the boundaries and done. We've been doing that. So just make sure you have that in mind. And there are different things out of this we can get. We can get the class width, class me um uh median, class interval, all of that, right? Okay. So we have ungroup data here. If I need to put it into this table, which could be a question that potentially come up, you'll be like, "Okay, one to five." 1 2 3 4 5 6 7 You cross it out. Seven. So, you add it in here and then you put the seven. Why is in the chat if you all understand this? I don't need to tell this is five in Italian. I You learn that when you're seven years old. Um, and you create you create your table like this. Now, most of y'all can see the mode here. What is the mode in this situation? If I just ask you what is the mode I give you even no context. I just say what is the mode you will see seven what you say it's not seven it's 1 to five. So if they give you access what is the mode without um context is 1 to five and usually they'll give the context they'll say what is the modal interval the modal interval is 1 to five the age interval 1 to 5 shows up seven times seven is not the mode the mode is also not four that's not that's not how it works right so we went through that over and over and over again I hope everybody finally has that no mistakes there as a silly one mark to lose and then you could be asked for the median And during the last crash course and during my class, I gave a nice little breakdown of how we view the median in this. So the median in this might actually be actually still in here. I could just watch and see. Most likely it's still no the median is actually dead in between the two because there's 10 here and 10 there. So the median is in between these two. So this is a kind of clumsy example to show the median. I'm going to show you it later on. Don't worry. Um, we'll just keep going forward to find the mean. This is where it gets good. There's a first column that you have to put in here called X. And X is what? What is X? What does X stand for? Come on people. What is X? Use the chat for a bit. X is the midpoint. X is the midpoint of each interval. So yeah, to get the midpoint is it usually turns out to be a like a 10.5 or 15.5. In this case, it's a three. The midpoint is 1 + 5 / two. So 6 / two is three. Why is in the chat if everybody understand that midpoint of this one will be eight 13. What will be the midpoint of this section here? What will the midpoint be there? Right here 18 and then 23. And then what next? We write fx. Does this mean anything with functions? No. Does it mean f ofx? No. No. No. No. No. This means literally f* x. So what number goes here? Yes, 21. Then 3 * 8 24. 4 * 13 that's 26 and 26. So that's 52. 4 * 18 I don't know what that is. 4 * 20 is 80 minus is it 72. And then we have 2 * 23 46. What's next? to get the midpoint of an interval at Gabriel. Um, you have stay locked in from now on. If somebody missed something that was done already, I put #lock in. This will help me remind people to stay locked in. You add that's why I have these two dots here and I I say 1 + 5. You don't see where you're going already. It's like you find the middle of these two numbers. 1 + 5 / 2 6 / 2 3 6 + 10 16 / 2 8 GT Gabriel. Then the sum of f(x) is what's next? The sum of x this symbol sigma means sum. So you add up all of this calculator people tell me what the answer is cuz you know you know 21 24 52 72 46. When you add up all of that you will get something. Additionally, you have to do the sum of f cuz these are the two things that will divide by each other. Oh god, why are you give me different answers? Eden is always a Eden. Oh, have you keeping what going on there? It lagging. It lagging. Okay. 73. If it lagging, listen announcement again. If your computer can't handle smoke, go on YouTube and take it in on the quiet end. Not this camera starting to follow me again. I didn't even pull up my hand, boy. So, what's the sum of f value? 20. They add it here as well. So, that's a good way to find the sum of f to just double check that you're doing the right thing. And the mean would be 215 / 20. 215 divided by 20. And you get your mean. We're going to see it in an example. Yes. Now, literally the means Xbar which is symbol for mean is sum of f(x) over the sum of f whatever that is 10.75. Congratulations to the people. Oh gosh. Right now them loss is real. Meanwhile, I had a Lenovo and it was painful. Determine class features for a given set of data. blah blah blah blah blah. We talked about this. We talked about this. Nonover overlapping intervals. We talked about that. Class limits. The limits is the numbers that given the boundaries is the actual far extent that it can go through go out to mention that. So the limits of the second class interval you can check it up with the stuff that we had here. Second class interval limit is six and 10. The boundaries will be 10.5 and 5.5. Yeah. Cool. Everybody, if you're feeling a little antsy, we have been on this, we have been grinding for the last 9 hours. Congratulations. 9 hours and 10 minutes. We heading up to the 12. So, take a little stretch. It's okay. You are human. It's okay to feel a little going for values. I mentioned that. Let's keep going. Class width. Somebody asked about the class width. The class width is the boundaries from each other. Not the Let me ask you here so you can visualize it. Lock in. Lock in everybody. The the class width is not 5 minus4. The class is 5.5 minus 0.5. The boundary distance excuse me. So it's actually five in this case. Yeah. Okay. So class bound class midpoint. Last thing I did this already. You add up the two numbers, divide by two. Whether it is the limit or the boundary, you'll still get the same thing. Construct statistical diagrams. So everybody take a stretch again. Statistical diagrams. Shayas commulative frequency is a huge topic, but the commumulative frequency graph comes under the statistical diagrams. Can you give me some example of statistical diagrams class? We talked about that just now. Ariana range is the highest value minus the lowest value. Maybe some of you all see some of these questions you're asking in past really worried about it. Range is the easiest thing they can ask you in a statistics question. Range is the easiest thing except maybe more they can ask you. What is the range? You look for the highest value and the minus the lowest value. So yes, we have pie charts which has been popular for like um the from 2010. Those of you who have been doing your past was 2010 to 2020. It was like the last time it was like the time was it was nice. It was real missing. But it has been showing up a lot recently for some reason. Last like five six years it showed up a lot. Bang bang bang bang real pie chart. So make sure you know to use the pie chart. The key concept in pie chart is what like what what you say one of the key concepts are in pie charts. There's no really trap there. You just have to know how to do it. You have to basically do the kind of find for one concept where yes the entire angle in a circle is 360. You have to find out how much one degree represent for whatever quantity. You also need to know how to use your protractor because the pie chart question is pretty practical. You actually do have to draw um sectors. So if you don't know how to use a projector, find out how to use it. Your projector is the semicircle thing with the angles. Is there anyone at this stage in the game has never used a projector before? You don't know how to measure off a You don't know how to measure off 45 Lea the duck. You do not use your protractor. A little lucky please protractor. Let me see if I could get a see if I can get a little crash course in protractor here. What is these type of diagrams again? PG transparent. See if I get you. Yeah, I think I get you. Copy image. Let's see if we can do this. I didn't want square paper. Fine. I ignore the squares on the paper. Well, at least the dog was brave enough to see five years of school and you never know. Oh my gosh, I really picked the ugliest projector, [Laughter] right? I really pick the most hideous projector on the internet by anyhow. So, Alia, your projector wouldn't have the ruler part underneath here, but you'll have this part, right? Focus on that. Look at your projector everybody. Can you do you not see that this is going 0 to 180 and this is going 0 to 180. Make sure you see on your projector otherwise carry back and dance some money. If you see 190 thing bootleg projector so 180 and let's say I want to measure this angle, right? Let's say I want to measure this angle. What I have to do, I don't know where the zero mark is on this projector cuz what you have to do is find the zero spot which is usually right here. But this projector even have it. This projector is for demonstrating purposes only. Um, just to show you I can do better than this. Let's give a little chance to redeem the name of math teachers out here because what in the world is that? Let me see if I can add a better projector. Come on. Come on. Boom. Right. I guess this projector don't have it on the two sides now. Just try one last time. Bang. I get through here. I get through the disc. Oh, the disc. Gosh. I really had the ADHD bad boy. So, yeah. Now this I I love this projector. I love this projector. Watch this. Watch this. Because most people protractor like this not. So watch your projector. See if this not look like the twin. What happened is that a lot of people use right here as where they putting the zero. I wonder if it'll move with the diagram. So like they will set it up like this. That's wrong. Do you see the center is here? So if your protractor doesn't have this little extra space, well then you could get this. But other than that, set your center on the center. What is the value of this angle? People give me what you looking like. What you looking like to you? Talk fast. This is not part of the crash course. We holding back progress here. It's looking like a 24 to be exact. A little less than the 25. But 25 is not bad. Why can't we say 156? Why we can't read it like this? Does this mean that was coming from this? If the angle was like this, then 156. If you say that you don't like your marks. So, let's do our next one since that was so much fun and we learned so much. What is the value of this angle? Now, see I line up my thing proper right on the elbow there. What he looking like? So this is a common mistake. So let's say it's I see a couple of people say 131. I need to talk about that. Let me just actually switch this line a little more. So let's say the line is more like more like Yes. So no no no no more like more like there. So what you call that the blue line 137 138 maybe even 139. Yeah. 139 139. Not 137. Was either 139 138. not 141 cuz some people is be counting this way and then all of a sudden decide to reverse count so they say 141 zoom in a little bit. I zooming in for the glasses committee. I just give my little chance to help out my glasses crew here. Um what about this one? This one is 136, not the 134. 134 would be here cuz 1331 32 33 34 Everybody understand? Now in uh in a pie chart question, doc, how about you? You catch that? In a pie chart question, let's say I need to plot a sector. They usually give you the first line. And I need to create a sector that let's say is 11 uh 15 degrees up. So what you need to do is bring your your protractor make it a little smaller. So you wish you could have do that in the exact bring rotate it around and you're trying to line him up to that to that line there people. That's the center point. So, what do you think we're going to do? We're trying to make a 150. Now, we're making him bigger so you can everybody can be on the same wavelength for the glasses crew. What's the next plan of action? We're trying to make a 115, folks. Lock in. So, if you're trying to make the 115, you need to find 115. Obviously, we're not going to find the black 115 because we didn't use a line across here. In fact, let me change the line so we could give the black the black measurements a little a little play. Right now, you using the blue measurements. So, let's say that this line was actually on this side. I don't know where the center is. Let's say it's about there. So, and we wanted to we wanted to make 115. So, we line him up. Bang. And you have duck. Where you do next? Lea, your hand up in the sky. You hit the 115 mark here. Now, you're supposed to just make a little point like this. Everybody cool? And then what you do, remove your projector. Try not to mark up the whole paper. Oh gosh. Oh gosh. You remove the projector. I'm trying to remove my projector right now, but real stress to get this to move. I was just now getting it. All right. Bang. You remove your projector. That was real stressful just now. And then you connect this. Now, obviously, you don't go all the way up to the dot by lining it up with the dot with your ruler. So, you use your ruler and you connect there. Why is in the chat if you understand this angle is now 115° everybody catch that? Nice. So now you know how to use your projector. Practice it now please. So when drawing the pie chart, use a ruler. Put a title. Insert the labels. Make sure you include the angle measurements. You're not required to shade the sectors for some reason. Some people just have to shade down pie charts. Right through going on who just sent me a random What? I can't even see where N I think that fake. I think that fake. Somebody just sent me a fake thing. I think that fake. I think that fake. This is the time all kind of sillinesses have people is be trying to send you fake um paper tools and things. So cricket. So you see how I have this labelled up. It's no m you see how I have this label up cricket angle stuff like that. That's what you need to do. Okay. Advantages of using a pie chart. Everybody look back in visual representation and they get an idea of the fraction with one single glance and keep it locking. Keep it locking. Keep it locking. Barraphs is one of the easiest statistical measures that statistical diagrams that come which is why it don't come. It's come right. So even use discrete data. The issue with bar graphs is that it can only be done using discrete data. What does discrete data mean? wise in the chat if you understand what discrete data means and if you don't discrete mean obvious tree okay my bad my bad my bad I didn't watch the paper name is a paper tree okay cool so y was a paper tree So discretet means like if I do this right and I say in this we have some red marbles and then we have some blue marbles we also have some black marbles that's discrete it could be counted I could individually count up those marbles and I could label this like this is my red this is my blue this is my you all understand black continuous data is the opposite Well, not the opposite, but I don't want to say antifist. The other thing then of of types of data continuous data cannot literally be counted. There's there's um a range of values and within the range of value any value can come. Infinite values can come. So things like weight, time, anything that you use that instrument to measure is continuous. It has to be measured. measured data is continuous data like weight or height. Um that's continuous data. So that's that type of data. We don't normally use a barraph for that. We use the cousin to barraph the big cousin which is which comes more regular as histogram. Yes, histogram. Very good Elijah. So with histogram instead of seeing like individual things like a red, blue, green or groups here, categories, you'll actually have what here? What goes on this axis? A number line. A number line will go there. So zero blah blah blah blah blah. And with histograms, what do we see with the columns? The columns actually touch. They join together. So histograms, the columns join together. So more on that later. Let's keep rolling. So this is a bar graph here. This came in in the notes. I did this with my students last year. Last year we do this people earlier this year. Take note of the you know earlier this year. Take note of the unit of yaxis in the top. Okay. Okay. How much how much what was the increase between 199? Wait. What was the question? Which which two years between which two years was the highest increase? I think that's the tricky question. Between which two years did we experience the highest increase? It was a CXE style question that came like this. Between which two years did we experience the highest increase? And a lot of people have 2000 as the answer or they have 2020 as the answer. Both of them are wrong. The correct answer is from 1999 to 2000. There's the highest increase. And how much did it increase by? How much did the accidents increase by? No. No. No. No, no, no, no. 20,000. Thank you, Kaya Bailey. Thousands of accidents, right? Increased by 20,000. So, that's the two tricks there. That's you ain't take you ain't claiming now on Elijah in the exam. You see it? Well, you watch out for that, right? Read the read read the labels. And that was the tip here. Y thousands of accidents. It happens in the exam. It happens. It's not like I'm just saying you have to take note of those trap cards right Z line graph just like how bar graph has its cousin histogram line graph has its cousin which is which is what FP frequency polygon well done Elijah is a man studying this thing by know the material. So the line graph is usually used why is in the chat and if you never do it why is if you know about a frequency polygon and if you never do that in your life before what happening in these schools in the Caribbean what is going on Lord so anyhow we'll talk about it soon. So the the the good thing is frequency polygon and histograms are not so um traumatic. You just need to do it once. You need to do it once. Uh right. So line graphs are usually used for stuff like this like years. It's usually used for a system of years like 1998, 1999. So we have it like this and that's your line graph. Cx has been known to kind of mix up the two the line graphs and the frequency polygon which is weird like to be really strict. This is your line graph. The frequency polygon is something else. A frequency polygon can work similar to a line graph, but technically it's supposed to drop down and make a whole polygon. Additionally, the frequency polygon uses the midpoint of class boundaries to create the frequency polygon. So, I have seen some cases where they call something that look like a line graph a frequency polygon when it shouldn't really be called a frequency polygon. Weird. All right. So, um histogram. So with the histogram, we use this group data or let's say interval data. You see the intervals here, ladies and gentlemen. So to create the histogram, what do you think we need? Do you think we need the midpoints, the upper class boundaries? What do you think we need? The midpoint or the boundaries? You're going to need the boundaries. You're going to need the boundaries. And usually just the upper class boundary. The only lower class boundary you need is the first one. So 0.5, 10.5, then 20.5, then 3.5. 40.5 50.5. So, let me give you the rundown quickly. Let me give you the plan quickly. Everybody lock in. Stay locked in. We We're running out of time. Got to finish this and the next thing in um two and a half hours and we still have bearings to do. Okay. So 10.5, 20.5, 30.5, 40.5, 50.5. Everybody watching, you're going to draw your axes. What do you think the highest value this will go up to would be? What's the highest value you think this this one is going to go up to? 12. This axis is always frequency or if you're doing a commulative frequency curve, commulative frequency. So this highest go this one go up to 12 ratio. You remember how you do this? So 12. So it start at zero six 1 2 3 4 5 6 7 8 9 however. Um and then this is the important part. Now we are going to show the diagram just now. I just show you how you'll size it up. You find 0.5 here. What do you think this is the highest that this one will go up to? What is the highest that the x-axis goes up to? 50. technically 50.5. So we want to make sure that we have a little room after the 50 50.5. It's usually just really minute. Some teachers teach you to draw it on the 50.5. I don't go through a lot of wireless 10 20 30 40. So what is going to happen is just right after the 10 mark the graph is going to be drawn and by the 0.5 the graph is going to be drawn as well. But I did this in class like two or three times. Like I literally showed you how to draw the graph. I went through the whole long process which might take like about 15 minutes. So I don't want to do it. It takes a while to do it on um online. But I want to see if you all remember what is the important tips that I gave you. This had nothing to do with touching the road. Liam that is for frequency. That is for communive frequency curve. This is a graph. This is a histogram we drawing here. What was the tip? What was the tip class in drawing this graph? What was the important tip? can't remember. I need you all to to challenge yourself to draw one histogram. Just draw one. A quick one. Even if you just draw a few columns to just make sure you you feel it out because overall using a ruler is not tip. Using a ruler is a must. If that tip for you like real adventurous so using that ruler is a mustby tip for this is to draw the vertical lines first. Let me show you why. This goes up to three. So you'll find three here. Let's say this is three here, right? Wherever three. And you're going to draw your vertical line up to three. And then don't draw the next vertical line up to three. So this 0.5, listen, lock in. This is important. This will help you be less stressful when you're drawing this. 0.5 goes up to three. The 10.5, let that go up to five. Like this. So the line that is on the 10.5 will go up to five. All right. 10.5. So the 30.5 or the 20.5, let's just print all the boundaries. Yes, 30.5. That's the lower cost bound I put in there. The 10.5, the 20.5, how much you think that will go up to 12. The 20.5, let that go up to the full 12. Now the 40.5 also let that go up to 12 because the next one coming down to seven. This one coming down to seven. this one coming out to two. Just a little tip will really help you. And then you just come in now and you draw this. Now, the reason why you're doing it like this is because this diagram have a tendency for students to have some scrappy looking on top of each other. Erase off this top line here. Bong, it just come out real bad. You have to do it to see why that's why that CV little tip is make a good difference in how you're coming up with the neatness and the quickness of your graph. So that's a nice tip there. Very practical. Um here they determine the upper class boundary and the lower class boundary for each group. Draw the x and y axis on the graph. Use a small dot to show like the boundaries for each category and draw the vertical lines first. Bang bang bang bang bang. Make sure everything looking good and then cut them off and you'll finish real fast. Do a couple do not even if you just do one before the exam cuz some of you all just skip over the graph revise it. Just do one before the exam and you should be good to go. Find a histogram somewhere and draw it. Cool. So look how this histogram came out. This is how it should come out. What are some of the key points you notice here to get the max? Because if they give you to draw a histogram, real marks flying like four, five, six marks one time on the graph. Scale, title, labels. You notice it's on the 10.5 on the 0.5. You must show the examiner that you're using the boundaries. You must show the examiner that you're using the boundaries. Everybody know how to do the scale. So like on this scale, I can see this 1 cm here stands for one person. I can see 1 cm here stands for 5 years. Moving on to draw your your frequency polygon to draw the histogram we need to use the boundaries to draw the frequency polygon. What do we need to use? Talk to me. All right. Just now in the next group you see just now somebody's asking something so this will be posted after yes for the terminated frequency curve if we have to draw the curve will the scale be provided or do we have to figure it out for the I read for the community if you have to join the curve but the provide no it could go either Sometime they give you the scale, sometime you have to figure it out. So make sure you know how to do the scales. Um yes, Elijah is correct. Elijah is the only man who revises where all the 10 20 answers I was getting from uh 12:00 where they gone. Yes. Yes. You have to use the midpoint. So the midpoint in this case you'll be adding the 1 and the 10 divide by 2. 1 + 10 divided by 2 5.5. 11 + 20 divid by 2 15.5. So you're using the midpoint. Zion, give me a chance, please. So you're going you're going to use the midpoints. And yeah, Elijah really Elijah, you have a student already. You're still waiting for it. I I don't get one already or you buy it. this man. Yeah, Elijah was going second. Yes. Wait. The man locked in right through. So, your frequency polygon going to come out looking like this. Ladies and gentlemen, are you okay with that? You're going to be using the midpoint, but technically you're supposed to close your frequency polygon by bringing it down to the 0.5 or on the 50.5 to make it a complete polygon. Elijah, how long you waiting on this? Let me just make sure them get you your message recently. Okay, good. Get you. Just let me know. Two [Music] months. No, they forget, bro. Something something happened. Message them again. Message them again. So, in the frequency polygon, we use the midpoint. The hoodie is expensive. Karen, we don't have the hoodie for sale or anything. So we don't mass produce it to lower the price. So it's some kind of 300 and something. I can't remember 100. It's not $500. Oh gosh. I had a name there. I'm not too sure about. I didn't check the list of that person on the list. You can check out V for me. If you hear any admin V V Ram I think V Ramdani. the RAM that yeah so this is how it will look let's keep going forward people determine measures of central tendency raw ungroup data determine what is most appropriate in mean median mode so measures of central tendency don't get fright when you see that the syllabus that just mean median mode um but you need to be able to determine mean median mode and other measures of central tendency like uh you don't need to know standard deviation where you need to know to talk about it but how to do it for ungrouped and grouped data Yeah. So the mode of a set of numbers is the number that occurs the most. Median means middle. You all know when I give this to class um people made a mistake. What is the mistake when this? If I give you a raw some raw numbers and I tell you to find the median. The most common mistake not forget forgetting to put it in order. forgetting to put it in ascending order. To get the mean, you add a whole and you divide. Now, we already established how we'll do the mean. If we have group data, you have to do that f(x) stuff, then find the sum sum of f and then divide." So, here's some ungrouped data. Please find the mode and the mean real fast. Real fast. Let's go. Mode and mean. Let's go quickly. I think this is from the notes actually. What's the mode? By now you should see that the mode is mode is three. Bang bang bang. And the mean you'll have to add up all of this and divide by how much? 1 2 3 4 5 6 7 8 9 10 11. Was it 2.7 something? Can't remember what we get for the mean for this when we do this in class. I'm going go and keep it here. 29 over 11 2.6 6 2.6 2.6. Well, Javante and Christine had some quick maths. Uh, before finding the median, make sure you put the it in order. Then cross out. Find the median. If you get two numbers in the middle, cuz you have to cross in pairs. Cross cross. There's one number in the middle. If there was two numbers in the middle, you add them up and divide by two. And that will be the median. So for group data, So for group data, remember you'll have to create a new column. The X here, what is the X? Now somebody beside Elijah, tell me what the X stands for. Midpoint. 1 + 10 is 11id 2. So you get 5.5. So you usually get 0.5 when you're doing the midpoint. This one is going to be 15.5, 25.5, 35.5. Then what's your next step? You had to get quick at this. Don't let it be the first time in the exam you doing this. Please multiply FX. So you get 5.5. 3 by 15.5 is I don't know what 3x 15.5 is. 46.5 46.5 3x 25.5. You know, you get 76.5 or whatever you get. And you keep going. You multiply. Then what's the next step afterwards? The sum of f(x), add up all these things, put it here. Add up all these things. Sum of f, put it there. So that's 14. And then you divide and you'll find the mean. Careful. Oh, look at that extra piece here. Who knew? Careful. So it's not 14. It's actually 14 + 6. The sum of f is 20. There's this extra column there. I don't know what going on in the notes. So modal interval is hey you did not reach boy. So you know we going to have a break just now. When is this break after statistics? After statistics but we still have some past papers to do. So the modal interval is 31 to 40. Let's look at this. 31 to 40 has seven. That's the mode. Lock in. Lock in. Um the mean we have to do that whole mean stuff. So let's see how it's done. Watch it everybody. Rate out of 10 how much you understand this 10. I know people in my class see me do this real plenty times. Sum of f sum of fx. See how we multiply across the stuff. Add it all up. Then the last step is to just divide this with this. Cooking a drum cook. Easy. Don't w it up. 32.5. What is the most common mistake for finding the mean? People will watch this. Add this up. Add up this this column. Get 20. I be like, add up that. Get 20 and divide by four. 1 2 3 4 5. Now divide by five and get four. So you're here make that mistake in class and you remember when you make that mistake. Do not do that. If you have to find the mean of group data, you have to do this fx thing. How you could tell me that the mean of the test scores is four when it have only one person get between 1 and 10. Everybody else get this person get between 11 to 20. Most people get between 31 to 40. So how you could say the mean? The average mark is four. The average mark is 32.5 and that's that's the only way you can find it. Relle appeal talk to me. This is for group data. Group data. Group data is when you see this indivi um interval stuff. Am I if I could share I would share. I'm a I'm a kind person but um it's online. You get get nothing. They don't we don't have that technology yet. So the median interval is the one in the middle. So I told you I'll talk about this. How do we find the median of group data? How do we find the median of group data? Fair play. Fair play. How do we find the median of group data? You have to be able to visual. You know what I'm talking about by how do you find the median of group data if the people them in here for a long while if the adins them remove you is because you just have your first name you're supposed to come in here with your whole name Liy surname or whatever I don't know have your full name rightsa andella um and well I know Liam from class. Back to what we trying to do. Median. Median mean middle. How do we find the median of this? Why isn't it? So somebody tell me n + 1 / 2 or half n + 1. test that is Q2 which is the second quartile represents the median. So it is the same idea. But how do you find the median here? How do you find that quartile? Well, you could use this idea a little bit and get to it. I'll show you just now. But I just want to show you this means we have in order one item here, three items here, another three items here, another seven persons here, and another six persons here. So we have one person age 1 to 10, three persons age 11 to 20, three persons age 21 to 30, seven persons age 31 to 40, and six persons age 41 to 50. How many of youall know what I'm planning to do now? So I want you to visualize like this is the lowest number. It will cross with the highest number. You seeing it? So we crossing like this class. Cross cross. And Marie and Marie talking to me or she might come on by accident. Cross. Cross cross. Cross cross. So the middle is going to be in the 31 to 40 region. The median interval is 31 to 40. Sometimes people don't get that on the first book. Like people just be like, "What in the world is happening?" Because you're overthinking it. So let me just show you what happened in here real fast. I think I showed it in the last crash course, didn't I? So in the last crash course, I was like, "Okay, let's do our own thing. Let's say we have we have two we have seven we have 16 we have 25 26 and 30 and keep keep it pumping. Let's not even say 30. Let's say we had another 26. And then we have a whole bunch in the 30s. 31, 32, another 32, a 39, uh 39, and the next 39, and then you have 43, 43, 44, another 44, and a 48. So let's say we want to find put this in categories. Now interval and we have 1 to 10. How many people we have 1 to 10 in the frequency? So this is the interval. This is the frequency. Talk to me fast people. In the 1 to 10 two and the 11 to 20, how much we have? Just one. And the 21 to 30. How much we have? One, two, three, four. Three. Three. In the 31 to 40. How much we have? 1 2 3 4 5 6 6. And in the last bit, how much we have? 41 to 50? Four. What is the upper class boundary of the second interval? By the way, let me just take a test on this. Upper class boundary of the second interval is what? 20.5. Good job. Are you cooking? What is the upper limit of the second to last class interval? What is the upper limit of the second to last class interval? 14. Well, you know what I'm saying? Why bigger body, right? We starting there. We locking back in then. We back on the inside. So, the limit is different than the boundary, right? Chantel lock in. Lock in. So, what what I'm trying to do is show you how we get the median. If if this was our data and we didn't have this table to get the median, it' be like cross cross 1 2 3 1 2 3 1 2 3 1 2 3 these two in the middle. So the median is 32 when I add up 32 32ide by 2 32 in if that makes sense. Am I making sense everybody? Or even if I had this one as 31 31 who asked that Shaniah 31 and 32 divide by 2 get 30 31.5 Shia I don't understand how we crossing out here. Oh, we just crossed the highest and lowest in. This is this is reminiscent of what we do, you know, when back in the good old days of primary school. The median, you order it up and then you cross cross. What happened? I get lazy. I get I I want to cross one one. So I cross three at a time. One, two, three, one, two, three. I cross maybe one, two, one, two, two in the middle. I can't say cross and take this one. I can't say cross and take this one cuz you always cross in terms of pairs. So since there's two in the middle 31 and 32 I can cross out the two right. So um add it up and divide by two we get 31.5. All right everybody know should know that that's basic stuff. You can do the same either here just we don't know what the numbers are. I know there are two numbers here four numbers here. You all seen where I'm going with it now? There was six numbers here. Three numbers here and one there. So, if I cross one of this, I'll cross one of this. I'll cross one of this, one of that. One of this, one of that. One of this, one of that. One of this, one of that. One of this, one of that. One of this, one of that. One of Oh, sorry. Well, that's what went on there. He was supposed to cross. So, I end up with a two down there. Class, it making sense? So, I end up with a two here. So I know I don't know what these numbers are because if I don't have this I don't know what the numbers are. However, I know that the median interval will occur between 31 to 40. Mel, ask a question. Ask a question. What you confused about? Come on. So you understand this part. and our number line x. So I just make up these numbers and I cross out and I get the middle term. You understand how we build this table? Okay. You know why this is two n? Why is this number two? Why is this number one? You're guessing. You're guessing. You're guessing. You're guessing. And your body off. You're wounded. You're wounded. Miguel, we have two numbers between 1 to 10. We have one number between 11 to 20. We have three numbers here. So pretend that this two come on. You use your imagination here with me just now. Just now. Don't say yeah. Yeah. I know. Understanding. So we have two dots here. These two dots represent that. You okay with that, Muel? And this represents this. So I just do down everything here because I'm pretending that I didn't see these numbers before. I just know I have two numbers between 1 to 10. So the same way how I cross this two, I will cross this with the 48 until I reach down into the middle. Bang bang bang. And then then it turned out that the middle is somewhere here between these two. So I know that it's here. It's the 31 to 40. That's how you find the median of group data in this nice way. In fact, when I'm doing it now, some of the some of the videos I saw on YouTube, I didn't even bother to put any dots. I was just like one four numbers cross here. So, I cross four numbers here, I have two numbers here remaining. I cross the next two numbers here, one remaining. I cross out this two. If I cross one number here, I cross out one in this. So, I have six numbers in the middle here. So I know that the median somewhere inside this mix. Instead of using the dots, I just use my imagination a little more by making. All right. So that's enough on median. I was saying that I would show you another way to do median. Well, it turns out that the median is given a special name. So let's just come up with some random numbers here. Zach, you better get a warning. M says Zach that is a threat. Zach, make sure and organize that one in. All right. So, one one 2 4 5 6 7 9 10. How much numbers are here? 1 2 3 4 5 6 7 8 9 10. Make it 11 numbers. So, boom. I didn't want the last number to be 11. I just want it to be 11 numbers. So, let's put 10 again. Okay. So what is the median? Somebody quickly tell me what the median is going to be. 1 2 3 4 5 6 7 8 9 10 11. Median is what? Isabelle, we doing we doing um statistics. Just how many numbers? 1 2 3 4 5 6 7 8 9 Are they getting 6.5? Are they crossing properly? Cross cross six. Six is the median. Six is the number in the middle. Now there's something that I spoke about here and you can see it happening again. There was 11 numbers but the median was the sixth number. Let's remove that. That is in the number six. That is wait to be five. The number is five. But it is in the what position? It's in the sixth position. 1 2 3 4 5 six position. If you have three numbers, 1 2 three numbers and it could be anything. I can have 21 30 and 31. What is the median number? 30. It's in the second position. So it's not 3 / 2 which is 2 which is 1.5. The median is 3 + 1 / 2. n + 1 / 2. n is the number of terms we have. So if we have 11 terms, this is the nth term. N + 1 / 2 or otherwise known as half n + 1. Same thing is how we find what the position the median is in. There are 11 terms here. Half of n + 1. Half of 11 + 1 is 6. The median is in the sixth position. So you count 1 2 3 4 5 6 and boom, you find the median is five. The median is the number five in this case, but it is in the sixth position. More on that just now. Let's go make it happen here. 21 20 + 1 / 2 or n + 1 / 2 or half n + 1 any way you want to look at it. 21 / 2 is how much class? 10.5 count 1 2 3 4 5 6 7 terms already gone. 1 2 3 in the 10th term here cuz we know the reason is 10.5 remember we had two numbers in the middle and we had to like when I pretended that we was doing it you have two numbers in the middle here somewhere that is where the median is that is me that is the other idea that you can use to find the median but it still carries the idea that you have to know that you count in each term here one term another three terms another three terms seven terms going on so The 10.5th term is somewhere in this interval. All right class, we have to move [Music] [Music] forward. We have to we have to move forward. So, let's keep it pumping. I have 3 hours to two hours to complete everything. It's getting and see people. It's getting and see. You should be able to estimate the measures for group data. Range, intercort, semi-ortal range. I feel I speed up the time when I teach it. Range, interal range, semi- Zach. So, how do we find range interal and semi-cortal range class? Good job, Anmary. Good job, Anmarie talking things. So, to find the range, you take like the highest value. It's hard to see in this. You take the highest value and you subtract the lowest value as well as there was something that I was mentioning that's coming. I trying to find it. Look at here. You all remember this. We just do this. So, I hope your memory shut. And I was talking about the median and I said the median is the sixth term. Everybody lock in. This term is called what in terms of quartiles? Q what? That is Q2. If I ignore Q2 and I just watch these what is the middle term in these 1 2 3 4 five? The middle term is two or it's in the third position. It's actually this one is called n + 1 half of it. This is a quarter of n + one. Remember n + one in this case n is 11. 11 terms + 1. 12 / 4 is the third. 1 2 3. The third tree RD third term this is called Q1 that's the lower quartile so we can already see that our upper quartile is going to be here you all starting to understand what quartiles is about I mean some of you know this inside out already this is actually our ninth term and we get nine because it's n + one 3/4 of it. So that's the what we call the quartiles. If we trying to find the interquartile range, the inter quartile range is and Mariana tell me what the inter quartile range is and no divide by two. If I divide by two, I'll be finding the SIQ, which is the right right is when you divide by two. So the inter quartile range in this case, what do you think it's going to be? And Marie, you want to know what country you're from? Guyana. I want the actual number. Tell me. Tell me. Whoa. No. Q3 minus the Q1. Q1 we have here. and Marie. So, it's 9 minus the two. Yeah, don't worry. When it turn when you turn on the mic and I talk in front of real people where things will start to happen, your brain is start to stop working. 9 minus 2. So, the answer is seven. Elijah and Javante gave the answer earlier on. The interquartile range in this case will be 9 - 2, which is 7. Now, when we ask interile range, we do we do not usually get it like this. This is raw ungroup data. We usually get it in terms of commutive frequency or some other thing. So let's keep moving. Keep moving. Keep moving. Keep moving. Somebody want to tell me why my graph looking half and half like this. Why? What page is this? 121. Why life can't just be stress free boy? Why we must have had a fight up in this thing called life boy 121. All I going to find is we have to just um see if I could organize him a little better. We have to do this communive frequency. We still have the whole of the big boy the last boss to do. They could guess what is bearings and circle geometry. The fun the fun topics the topic that you love. Topics that you can't live without. Look my commulative frequency curve behaving itself normally. So why it had to come out? Oh gosh. Yes. This is a little pH break here. Bang. Now, watch out for this graph, right? They're doing some I don't know if you hear still. They're doing some weird things. Let me see if I just could save myself by exporting this document again. I want to store as a PDF version two. You always do it when you have a new thing. Version two. Version two final. Version 2.3 final. Final. Real. Really mean it this time. Um, supposed to be making your way downtown. See if I get through here. Let's try to get this graph up for you. Now I have to scroll all the way to page [Laughter] 121. So you should hang around. You should hang around. 121. We coming into some important stuff here and I'll show you how to collect some marks and bearings and stuff. Someone I know you're trying to choose your peace, trying to protect your peace. We should hang around. Come into some nice stuff just now. That was actually funny to read. Um determine the median of community frequency below. So if we have to determine the median Which axis are we looking at? The y axis or the x axis? Are you don't believe everything I see on the internet? Somebody say random things on the internet. Are you going to believe it? We're looking at the y-axis. So the y axis and how do we find the median? Again, the Q2 is over here. 60 60 + 1 divid by 2 is 30.5. So it's basically just above the 30 line or on the 30 line is good enough. And the median isn't 30. The median is actually 20. So the median is found on the 30.5 value which is which works out to be the value 20. 20 [Music] years. And do you know how we'll find the interal range? Now everybody, how do we find the interile range? Well, this is Q3 and this is Q1. Q3 is 3/4 away from 0 to 60. 3/4 of 0 to 60 and 3/4 of 60 + 1. Let's just use 60 for time sake. It turns out to be around 45 45.75 or something like that. If I use 60 plus one. So the 45 mark is where I'll draw my this is the touch the road thing. Touch your own. Come down read off this. This is 21 22 23 24 25. So yes this is 23. Similarly you do the same thing for Q1. You get 17 23 minus 17 and you'll get your interportile range. 23 years 17 subtract to get your interortile range examples using raw data. Can we just quickly find Q1, Q2, and Q3 here? Quickly, everybody, I'm going to take a a seat. So, I'll stand up maybe for the last hour. Quickly find Q1, Q2, and Q3 here. More snacks provided to by Pringles. So some tool day cupcakes and thing really going on like this like this lady um buy some Amazon package some kind of thing something I brace myself. [Music] Wow, that was fast. I mean, if you understand the idea, you can see that the median is 41 and then you get the median of these. The bottom half, that's Q1 and that's Q3. How much items you have? One, two, three, four, five, six, seven. So Q1 is found on the quarter of n + one a quarter of 7 + 1 a quarter of eight is two the second not second term the second term so the second term is 13 similarly you could do the same thing for the rest of them and the IQR here Anna the IQR is going to or Anmarie is going to be Q3 minus Q1 which is 60 - 13 which is which is whatever that is I don't know 47 sha right so how that how that looking you all understand Q1 Q2 and Q3 now some of you all understood it from before a final little idea on on statistics Before we go into some past papers, standard deviation, I spoke about this. This shows up. All right, Leela, you have to come through. Ra, you have a question. Q1 is the formulas remain the same. Q1 is a quarter of all the tables + one. Q2 is a half of N + 1 and Q3 is 3/4. It's like you dividing the data into quarters. Standard deviation in maths only requires you to understand what it means and these are the key words spread deviates um mean. It really talks about how much the data varies from the average mark. So the average mark of these this height of building is whatever you see how much it varies. Then let's go look at this one. The average of these buildings only this one really varies far from the mean. So this one has a lower standard deviation than that. Standard deviation speaks to how much the individual values on average vary from the mean. The deviation or the spread of the data from the mean. So some uh uh If we have two data groups of data and both of them the mean which we call call Xbar turn out to be the same thing let's say turn out to be 12 but the standard deviation which is sometime called SD or sigma the standard deviation for this is seven and the standard deviation for this is 18 what this tells us is that all the individual values in this group of data Let's say A and B. The group of values in A would be a averaging around 12. Let's say this is 12. But the it only varying on average like about seven away from the mean. So you'll see values like 20 and you shouldn't see too high values. This one it averages around 12 but you there are some larger variation. Why is in the chat if you all understand what variation is talking about? So the key thing here is to take away the words that I'm saying deviates and spreads further from the mean because that's all they ask you. That's literally all they ask you in the exam. And here it is. Read this. Memorize it. Watch how I use the keywords. It's on page 119 or so. Draw cumulative frequency curve using continuous data is um the running total of frequencies. Let's go check it out. Oh god, my graph going wild again. Y just give me one second. Take a look at this and see if you understand this in the meanwhile while you're watching the tips. If I can just sort this back out once and for all. Why is it doing that? All right, I think get back on track there. C match crash course 12 hour. Save this session. Session three version three. I want this to save as a PDF. Let me just see if I can bring it up. I'll have to scroll all the way back to page 120. How you feeling class? Everybody good? Yeah boy. If you know hype towers after that win cuz everybody else saying that the world is going on with come on be real. I watched that match in Miami. I was running all over the hotel room like oh god oh head spinning watermelon in hand. And I get my food. So I good thing. Tired but going still. Golden State Warriors. This man say them not beating Ma. What really going on with you Pablo? Two different sports, two different continents. Lock in Pablo. Lock in. All right. So draw unless you're talking to somebody else. Maybe I missed something. Maybe I missed something. Draw communive frequency curve. Excuse me. So let's say we have this. Why is in the chat if you understand how these values come in? Rachel asked some questions and I not seen it. That's not I don't know what you mean by others. You're multiplying frequency curve. You add the FX thing. Yes, you're multiplying this. So you have two here. 2 and 9 is 11. 11 and 25 is 36. That's how it works, people. you keep adding the last value together. So those of you who type N, that's communive frequency. If you type N and you never do communive frequency before, some strange things has happened. I don't know why you never do commulative frequency before. It's a little upsetting actually because you have the whole year and even if your teacher didn't do it, you can look it up. You can watch the syllabus. It's a key compon. If you do one pass paper, you see communive frequency. If I if you do one pass, you see commumulative frequencies come like 80% of the time. The most popular thing to come for statistics is commumulative frequency. So not knowing how to do cumulative frequency or at least never tried it before two days before the exam is bad and a lot of people part including yourself. So 2 and 9 is 11 and 25 is 36 36 and 30 66 66 and 106 and this number needs to add up to how much people in the survey. So that this means there are 80 people all together class. You okay with that? So I also included the upper class boundary. Why is that? So because the upper class boundary is what we use to draw the communive frequency curves. So 10.5 20.5. Why do we use the upper class boundary? Is the highest possible value that these two people can be in. There are nine people here. The highest possible value that they can have is 20.5. So if let's go to the graph. This is 10.5 mapped up to the two. So what it's saying is that two people exist. Two persons exist within this region. You all understand that? Even if you don't understand, you need to just use those values. The 10.5 maps up with the two. The 20.5 mapped up with the next commumulative frequency. Not the nine but the 11 class. So that's why we get the commulative frequency. And since this value just keeps going up and up, it will never go down. It usually for for natural distribution, normal distribution, it'll form a curve that looks like an S. And that's called a O G I V O G I D E. So the touch the road you touch the road idea is like if you need to find the median, there are 80 terms here. The median is going to be 40.5. You go across here, touch the road, it's happening on the 40.5 value and then you come down. That's what Liam is talking about. So the median turns out to be 31 32. Class, are you with me? Leila, what was your question? Yeah. Speak up. talk from a chest. We'll talk about all them predictions and thing at the ending of class because that could go into a long discussion. So um what's what else somebody asks? Yes, the boundaries are where we use to make the points. So we use the boundaries and the points. The boundaries is like the x values. So this is like your x coordinate and that's like your ycoordinate. Let's say let's see if you could find this point 50.5 and 76. 50.5 mapped up with the 76. You seen it? That's 50.5 76. Okay. So from this you can do all your Q1 stuff. You can do all your IQR things. Just remember it's not the values on here. The values here. We'll see it in some past papers. We'll see it in some past papers. So the last part is probability. Now usually this is all I teach in probability for the last 20 years. That is what has been coming in probability. I will haven't been teaching for 20 years. Oh my god. I have been teaching for 20 years. That's insane. All right. So the pro so the probability is usually I I started teaching one time right out out of school. As soon as I finish um form five, I actually taught in form five. So the number of desired outcomes divided by the total number of outcomes is usually what the probability is about. So like if I give you a a box and there's some x's and then there's some whatever that is um squares 1 2 3 4 5 as I told you what's the probability of pulling a square. Well, the desired outcomes, there are two of them. Total outcomes are eight. Two out of eight. So, 1/8 class, you all understand that or 12.5%. So, it could be expressed as a fraction of percentage a decimal [Music] 0.125. Um, that's privately that's not like in a government school or anything. So, using communive frequency alone, you can come up with probability ideas here, you know. So like like with this what is the probability of someone being more than 60 years old? Well first you have to find the 60 year mark or they don't have it more than 40 years old. You find the 40ear mark you run up to the touch the road Liam bang you touch the road you come across you see 65. What that number means? That number means listen listen everybody that number means we have 65 persons who are under the age of 40. Why is in the chat if you understand that I say 18 is supposed to be a quarter sometimes I short circuit when I talking. So therefore it'll be 12 25% and 0.25. My bad. That's just to see if I paying attention. So this is 65. Um what I did I asked the question how much people what's the probability of picking somebody who is over the age of 40. So it turns out there are 65 people under the age of 40. The survey is out of 80 persons. So can you see that there are 15 persons class? Y you'll get that there are 15 persons over the age of 40. That will take me up to the 80 mark. And then I could work out the probability like saying 15 out of 80 alto together. And you all could reduce that for me. Five and five into that is 116. However, correct? So that's your probability there. So let's keep it pumping. Let's keep it pumping, ladies and gentlemen. Now I was saying usually statistics probability is like that. Um there's an example here in the notes you have to take a look at it. 76 was the number instead and it turned out to be only four persons. Um but probability has been showing up with probability space. Anybody see the probability space questions or take a look at it. I think I did a video on it. The probability space questions. So let's just quickly talk about that. Oh my gosh. I only have an hour and a half and I teach bearings still. I tell the time running out. So the probability space questions will look like this. Let's say we have a dice. How much sides on a dice people? One, two, three. I think I talked about it in the last scratch course. So to do the diagram. Yeah. And I mentioned ADMAT students need to know to do inside out. One, two, three, four. Admat students see this all the time. One, two, three, four, five, six. So, one, two, three, four, five, six. Or sometimes you write it the other way around. You may see this, see it like this in the exam. One, two, three, four, five, six. These are the possible options. And die one. Singular for dice is die. Die two. That don't mean anything bad people. One, two, three, four, five, six. And class. Lock in. lock in the two items you can see here is one one or one two or one three this would be 21 in the chat if you understand what's happening here and if you don't understand This some people don't understand it. That person was trying to be funny. I miss a joke there. That person making a joke. I the only man making jokes here. You confuse people not realize that joke he was trying to make. But you confuse me. Anyhow, so one one and um this is like coordinates in this is like coordinates. So we have one map up with the one, one map up with the two, one map up with the three, one map up with the four. You all seen it? It's almost like the coordinates Relle Lea. Yes. It's like So what what would go here then? Four. Four. What would go here? 52. Now, there's some occasions where they'll put an operation here. Let's say you play Monopoly. There's an operation here. Instead, if you roll a three and a two, you don't write three, two. You're going to write that's five. I'm adding the number. So, instead of having two two here, I have four. You want to understand the difference here, class? Five. AJ you're looking. So let's do it that way now. So I showed you one way. There is the way we could do a operation here. Um I think in January or if it's not January is last year there was a multiplication here. But let's do the addition. So this is two. This is three. Three. Then we can see now these numbers mapping out. Five five5. And I was saying I think in the crash course I did this and I was saying that this is why seven is the luckiest number if you have two dice. You all watching this pay attention everybody. David you're on your last string there right? So keep it keep it nice David. So seven is the one that show up the most amount of time. I did a video on it. All right. So let's cousin. What going on cousin? So 6 six six six let me not say that too much time 5 five and then here we'll have four four three three and we have a nice two there if you rule no sorry I'm doing this wrong this one is not going to do in reverse this one is going to go up again so I'm going to get six and two is eight then I'm going to get nines or you can check out the video that I did on this tens 11s and twelves Celeste is not family actually but some kind of cousin. This is the same thing Rachel we doing um probability. So then I could ask you what is the probability of rolling a 11? What is the probability of rolling a 11? Well, it's two two 11s out of how much is the whole sample space? 36, which is 118th. There's a 1 in 18 chance that you can roll 11. There's also a 1 in 18 chance that you could roll what? Three. The probability of rolling a three. There's also two threes out of the 36. Is that making That's not [Music] that. All right. Yeah. Hi I no. Yes 36. You get it there. You get it there. All right. So yeah, Jimmy was blocked. So, that's the person I letting I leave the admin still letting people. Y'all, if you're if you're um if you get knocked out, you're going to have to use the YouTube cuz we're getting a couple people. So, I can't just letting people randomly who thinking that they have the link. I have to let the admins let you. So, if you get locked out, you have to use the YouTube, right? Unless I identify your name quickly. So try and stay logged in. So let me just ask a quick question on this before we go because this is this has been train up recently. What is the probability of like maybe rolling a nine? What is the probability of rolling a nine? out [Music] of 36 which will break down to one night. Making sense to everybody now. So, they want you to be able to identify a sample space, get the total for the ones that you want, like the desired outcomes, and put it over the whole sample space and work out your probabilities. Um, for example, a coin has heads or tail. What is the chance of flipping heads? Well, there's only one out of two, 50%. A half. A dice. If it's just one dice, what's the chance of rolling a five? one out of six. There's only one chance of getting the five out of the six different options. What is the probability of rolling an even number? This is normally how it's come. Probability of rolling an even number. There are three out of six. A half probability they're even. All right. Okay. Let's keep rolling. Let's keep going. This is just how playing cards are distributed. This contingency table could show up in the um multiple choice. What is the probability of selecting a female at random? David is one of them two. That's very interesting. that same person I talked to earlier. What is the probability? Y'all answer the question. Don't get distracted. What What is the probability of selecting? So it's actually just one person with multiple emails. So it seemed like it's actually a person from class using So the person still here. A person from class using multiple emails. Wow. Anyhow, so the probability is 12 out of No. Probability of selecting a female is what? Right. So 48 over the 100. And is this breaking down to the 12? That's why you're getting 12 out of 25. All right. Good job. 12 out of 25. Um four into this is 12. Four into this is 25. What is the probability of randomly selecting a male that prefers cats? No, no, no, no. A male that prefers cats is 10. There are only 10 males that cat 10 out of the total. So 10 out of 100. Now there's something in ADMATS called conditional probability. There's something in ADMATS called conditional probability. So if conditional probability works like this. If and I'm I'm mentioning it because I'm not too sure if they will give you something like this in the multiple choice. What is the probability that given you selected a person who is male, given that you selected a male, they will prefer cats. That's 10 out of 52 males. You all understand the difference? So if you already know, if they say given that or you already select a male and you want the probability that that person likes cats, it'll be 10 out of this new probability space cuz the probability space is now 52. Emma, it's a it's you have to go back to the simple idea, Emma, like with the X's and the boxes. What is the probability of picking a box here? There are two out of six m. Are you good at that? Which could break down to one/3. It can be expressed as a fraction, a percentage or decimal that say 0.33 recurring 33 and a third%. So the probability is 1/3 year and arrow. I just said you could leave it as anything. You could leave it as a fraction or a percentage. Usually we just stop as fraction unless they request. So what is the probability of randomly selecting a male that prefers cats? This case is 10 out of the entire total you selected of this person male cat male like cats and out of um 100. So it's 10 out of 100. Given that female is selected, what is the probability that she likes dogs? This is going to be Oh, look at look at here. Given that. So what is the answer for this one? Yes. Nine out of the 48. I didn't even watch the next question. The next question has that idea. So let's keep rolling. We have to go on into the other stuff. [Music] Um two question seven. Uh seeing a new person trying to join here. Let me just double check this in the meanwhile. Why y'all to take a look at this table? One second. So yeah, so if we take a look at the uh this um we have to do some commulative frequency. Let's go. Let's run through this real fast. The next thing here is going to be how much? 30 then 46 then 58 then good job Luke 66. Copy and complete the table using a scale. So I'm going to show you how this is drawn to draw this. What what other column would I need to add in? I would need to add in what column here. Jonathan, why you say FX? FX is only if you're working out the mean? Yes, the upper class boundary ratio. Jonathan, I looking for you there. So the upper class boundary, what assumption you have made in the in drawing the curve through the 0. You have made that there's the assum not zero supposed to be 0.5 anyhow and that's not even that no one gets below zero that no one gets below zero. Yes, Zach. Send it. No problem. Uh let's keep going. The pass mark is 47. Use your graph to determine the number of students who pass the test. So we'll see that in the graph and then there's a probability question. So let's go quickly to the answers. Correct on this. Look, you have a question. That's that's what I supposed. So 0 is not where it's supposed to go. It's supposed to actually go to 0.5 according to lower class boundary. But they ask 0 0 is a very good question actually. I mean you understand real things. So this is how the communive frequency um looks. This is how the communive frequency would look. Notice how we use the upper class boundary. We went through this already. These lines were probably for the um the probability question which was what was it? What is the probability that a student chosen at random had a mag less than or equal to 30? And then there's one with 47 number of students who passed. So they run from the 47 touch the road Liam from the 47 it cracked 40. So that means 40 students got under 47 and the remaining students got over 47. So if there were 70 students in all, how many pass class? [Music] Just one second. Yes. Yes. So, it's going to be 30 because the top the top amount of people is 70. I just blocking a couple people. So, 70 minus 40 you get 30. 30 persons would have gotten above the mark. Well done. No student obtained a mark of zero. Zero at zero. If you draw zero at zero, no student. Well, they say if you draw the curve through then, but really it's supposed to draw through 0.5. That's the lower class boundary. According to this, the pass mark is 47. Then we have the 30. Um, what is the probability that student chosen at random had a mark less than or equal to 30? Well, we just find the 30. We go up. We have 15. How many students scored under 30? 15. What is the probability? 15 out of the total class. You'll see now the probability usually comes. Is this making sense? All right, let me I'm going to um One second. So that is the same email again. Now I know I used to get a little trouble in school but that's a um that's something doing this generation like you will create four different emails and then go in a matt crash course. Okay let me just keep going. So, so um each of each of 75 males recorded the name of I let in that person just now I was checking the list and your fellow just checking the list recorded the name of her favorite sport the number of girls who chose track and cricket as shown on the bar chart below. So this is a barraph barraphs come easy. How more How many more girls choose cricket than track as their favorite sport? Make a next email. How many more girls cricket than their favorite sport? Yeah. Yeah. I I'm I'm not allowing new persons again. So, it's going to go from 12 to what is this number? How much are they getting there? Five. If this is 18 and that's midway, this is like 17. So, 12 to 17. Five. And it worked out pretty fast. So, five. Five. And so bar charts is pretty easy. I don't think I need to spend too much time on this. I want to skip to the next one. Is that okay? Everybody understand how they'll get the mood and if they tell you it to draw next bar that's that's straightforward stuff. We're going to go keep on the tougher things, right? The solutions will be there too if you need to watch it over. Um the next topic is yes. Praise the Lord. So we have an hour and 45 minutes to do geometry and trigonometry. So where which pH am I going to? So because this topic is usually super hard shouldn't be that it should everybody should be as lock should be as locked in as possible right just give me one chance. Yep. So right now we are going into geometry and trigonometry section 8 which is on page 221 thereabouts. So everybody take a deep breath. You getting two minutes to catch your breath here before we go on to the 200 page 221. I was planning to finish a little earlier but it looked like they're happening. might finish later. All right, so geometry and tricks. Ken, what is up? One more minute. Let's take a little one minute here. Maybe see if I could wolf down something. All right. So, we're getting ready to go back in. For those of you watching the recording, sorry for eating in between is a long crash course. I tried to keep my energy high. Um, somebody on YouTube, right? I'll leave for one second and doing my admats now. All right. Jada Ramsey. Y'all, if you're getting knocked out, you have to use the YouTube, right? There's this one person who is um lonely tonight who's jumping in the chat regular. So, the YouTube link is for people to put in real comments. Okay. So, points, let's go into geometry and trigonometry. points are stated in the form of XY everybody know that that's like coordinate geometry everybody locked in I'm turn this off end oh my gosh I think end session right lines straight lines infinite oh we don't need to learn all of this parallel intersecting perpendicular meter 90° intersecting just intersect line segments is part of a line similar array. But in array, there's a start and there's no ending. Um, in a line segment, there's a start and an end. Curves, a curve is a line that is not straight. No. Planes, um, times, angles. These are some definitions that you all can run through to just get into the idea of things. We can have stuff in geometry like um, on a cube, we call these things the vertex. Everybody know that? Luck in guys. We have the vertex. We have the face. And this piece here is called a edge. So you should know how many edges, faces, vertex and these objects. And then we start to get into the good stuff that would lead into excuse me bearings and you know the tougher topics when two parallel lines usually the parallel lines that we encounter in the exams will be the north pole. Why is in the chat if you know that you're doing your past papers here the documents will be posted. So you have the north and then you have a transversal. You have the north and you have a transversal and this angle and that angle will be the same. They are called corresponding. This angle will be the same with this angle. That's sorry it adds up to 180. So these are called coerior. The same angle. If I compare to this it's it's alternate. So those are the same. So essentially it creates a situation where you have this angle being the same as everything I'm highlighting in blue here and then we have vertically opposite and all of that stuff. So it already posted very nice. So here we have the corresponding alternate vertically opposite coin and these come in vital. These are extremely important for um solving questions in bearings. need to know to construct lines, angles and polygons and use using appropriate instruments. So let's just go through some of the ideas to bisect a line. Let's say we have a line. Why is in the chat you know to bisect it? Yes, we do that today. Do you know how to bisect it? You'll go like this and like this and you come down in between. And that's the bicep. Now, how do we get these two arcs? You stick your compass in there. The compass is the thing that looks like this with the pencil in it, right? And the sharp point goes here. Actually to draw circles and thing and not only to chuck your friend and tell them. So you put this in here, open it out and you just draw a arc like this and you a little more than half. Very good question. So you kind of estimate where half is. You could even use a ruler and find a half and just open it a little more than half. Open it, draw an arc, stick it back here. You need to keep the arc the same length. So once you draw that first half, you can't play with it. You can't change length. You can't change the diameter of the radius of that back. You connect these two points here. Draw a line straight down. And that's your bisected. So that's the first thing that you need to do. If you all need to see me actually do it on a piece of paper. How many of you all watch the video on construction on YouTube? You'll watch it. Nice. Go and watch it if you haven't watched it. It'll kind of break it down to you. Even if you like my boy Elijah never see a gym, she said Elijah say. So you should also know how to biseect an angle. Let's say we have a given angle and we have to bisect the angle. You stick your compass in here. The pointy part sticks there and you draw the arc like this. Now what next? Keep it the same length. Usually keep it the same length. Stick it in there and draw a next arc. Now you have to make sure it's the same length for this part. You keep in the compass and you're going to make a next arc up here. Bang. where the two intersect you have just bicted the angle elementary things. So those two ideas are very important in transformation. Those kind of bicting the line ideas. You saw it in the last crash course when I did it. How we do that transformation stuff. The reflection or we finding whoever maybe miss talked to you a little bit about reflections and things earlier on. Um how do we make a 30° a 45 degrees a 60° a 90° 120? Out of all of these, which one is the easiest to make? If they ask you in the exam to construct a 60°, no. No. To show on a construction 90 is not the easiest to construct, you know. To construct a 90, you could bisect a line, but usually that's not the full construction that we do for 90. So, the easiest to make is actually the 60. Those of you who do the technical joining thing all your ears know from a distance. If I want to make a 60° here, all I have to do is open my compass, draw an arc, stick my compass in here, keep it the same distance, draw an arc, and connect that. I would have just made 60°. As simple as that. I I don't want to go through this too in detail because this is a whole two hours, right? So, watch the video on it. I need to show all it's it's it's for some of you all who never watch it. Let me show you. Let me show you. I'm going show you to give you a little motivation. Constru Hey, Coffee come out with that new song. Why construction curing Springer? Hey, Grand coming out or something. I seen that. And let me just minimize this way. This person wasn't here again in a next thing. I don't understand. This is 2020. This is 2025. So this man l all he So you see it. So you just click on this and you could um go through it like I show you how to bisect the line, how to draw the line with um compass. Y'all seeing it? So, please go watch all this basic [Music] stuff. All right. So, bicting um drawing 90° all of these stuff. Make sure you know how to do it. The usual idea that will come is that they may ask you to draw a triangle. They may give you a length of the triangle. They may give you uh Alex your camera. Come on, Alex Andrews. They may give you Alex Andrews, your camera. Come on. They may give you, let's say they you have to create a triangle that is 60°, 7 cm here, 60° here, triangle A, B, C, and they also told you this length is 8 cm. So you draw a sketch, then you come here, draw your baseline, you open out, you draw seven, then you come here, do your 60°. As I said, check out check out the video on this so you can see like me actually doing something on this. Then what you need to do is measure 8 cm on your compass and come and make an arc there. It turn out that this this might actually come out at the right angle looking a little looking. It wouldn't be 90° there. So, I just do something really fast. Let's keep going on till we get into the good stuff. Here's an example of a construction. In the notes, XY is 7.5. They give you an angle and they give you a next side. And they also told you one is 90° or something, but here it is. Identify the types of symmetry. We have line symmetry and rotational symmetry. Now, this is something that very rare. Alia, stop complaining. Keep locked in. So, something that's very rare is rotational symmetry. They might ask you how much rotational symmetry an object has. And this showed up in a couple paper twos ever. It's rare though. It's rare that it can show. Why isn't it you have an idea what rotational symmetry means? So you could go and research it if you want. This object here has two lines of symmetry. That's your normal type of symmetry. The rotational symmetry is how much times you could spin it around. Let's say I have A B C D. It needs to go in a circle. A B C D. How much times you could spin it around and it will go in its same shape without the points reoccurring again? So there is this um idea where if it can only do it I run out of energy for myself. If it can only do it. All right, let me show you what I mean. If this turns around like this, can you visualize that B will come over here, C will be here, A will be here, and D will be there. That class, that making sense to you all? It turns around like this. So because of that, the rotational symmetry that it will have is one. So that's what they're talking about when they say rotational symmetry. If something has no rotational symmetry, that means it can only exist in that one spatial arrangement one time. Like maybe an irregular solid would have one rotational symmetry. I can only get it to look like this once. So rotational symmetry is the type of symmetry occurs when a shape can be rotating around a fixed point. Order of rotational symmetry. I have the information here for you to check out. solve geometric problems using lines, angles, polygons, congruent triangles, similar figures, faces, edges, solids, classes of solids. Let's just skip through that real fast. Now you see they don't really bring the primary school they don't really bring the primary school aspect of it where they ask you stuff about rumbus and thing but it could show there is like if you those of you who do the past people you remember that that came in a question three a time or question yeah I think a question three and they asked you to finish off the table and there was rhombus and trapezium and stuff like that right so you still need to know what are the CXC some of CXE favorite shapes out of these rhombus and trapezium. So rhombus and trapezium. Make sure you know them fellas kite show up here. Make sure you know what is a rhombus. A rhombus is a square that has been squeezed. So all the sides are equal. The opposite angles are equal. What is a trapezium? A trapezium. There are like three types of trapezium. There's a right angle trapezium. What makes a trapezium is one pair of parallel sides a quadrilateral. All of these are quadrilateral. There are triangles and there cases for congrent. I'll talk about this word just now. There's right angle trapezium. There's the isosles trapezium and then of course there's like a scaline trapezium but you still have the parallel sides in all of them. Congruency means that you could take the shape and it will like like maybe this triangle is congruent with this triangle. It's the exact same triangle. Turn it a little bit. I can take the triangle and cover the next triangle and there would be no overlap. That's congruent. However, there's another word that goes hand inhand with congruency. Who can tell me what word I'm about to say here? It starts with S. Similar similar so similar triangles have the same angular arrangement. So the angles map up to be the same but they are different sides. Making sense class. So these triangles are similar and there are some rules serve around them. The angles will be the same but this might be two. This might be four. So if this is two and this is four and this is three. But we know this is six. solid not drawn to scale or whatever. So they related by a scala factor. Similar triangles congruent means it's the exact same thing. Properties of similar triangles. I just mentioned that there. Um so let's keep going. We're going to see some we're going to see all of this in past papers rather than just bore you with the theory here because this is just stuff to read. Everybody know a square, lines of symmetry, rectangle, all of that. Pay special attention to the rumbus. This rhombus is drawn that it looks almost exactly like a square, but a rhombus could be squeezed in as well on both ends. A rhombus could be squeezed in like this. A kite is something that looks more like this. Like a kite, like literally a kite. There it is. Kite, parallelogram, trapezoid, trapezium. We normally use the word zoid and stuff when we're dealing with solids like cuboid and cube. um supplementary angles add up to 180. That's the word when something add up to adds up to 180. So all the angles in a triangle can be called supplementary. Although we use the word supplementary normally only talking about two things. So but it's not unusual to see talk about three things. I can say if this angle is X and this angle is 50 and this angle is 70, you all should be able to tell me what the value of X is. What is the value of x class? Lock in. Lock in. What is the value of x? 1 hour to go. 70 and 50 is 120. So the value of x is 60°. Cuz all these angles here are supplementary. Angle measures the degree of turn. Everybody know acute means less than 90 more than zero. That's acute. Obtuse is when it crosses the 90 mark between 90 and 180. That's obtuse. Reflex is when it goes past 180. Reflex is when it goes past 180. but less than 360. There we go. Examples. Um, we have arc chord segment. We have arc, we have chord, we have a segment, arc, a chord, a segment. An arc is just piece of a circumference, part of a circumference of the circle. You already know the circle terms and thing coming up shortly. Circle theorems coming up. And you're spelling the wrong ly. Um the corn is a straight line that connects with the circumference of a circle. One point on the circumference with the next point on the circumference. AB is a chord. A segment is what happens when you have the chord. The chord actually splits the um circle into two equal pieces P Q. A sector is bounded by two rad and an arc. Two radius arc make the sector different than segment. They will tell you I don't understand the question. So that's a segment. This is the sector. tangent. Well, of course tangent, we have the normal, we have the tangent. Tangent is a line that touches the circle once and can only touch the circle once. Let's move right into circle theorems. The first theorem says that the angle in a semicircle is always a right angle. So if I gave you this triangle and I told you to find the value of angle O P QR. I told you to find the value of angle QPR. What is it going to be? angle QPR is going to be 30°. And the reason they get 30° is because we now know that this has to be a right angle cuz it's in the semicircle. So, it makes a right angle triangle there. The second theorem says that the angle at the circumference is half the angle at the center. So, let's do a next quick question. If I give you um this I said let me see let me name is OPQI R and I said R RPQ is angle RPQ is 40°. What is the value of R O Q? What is the value of O RQ? Come up with some of them there. So RPQ is going to be 80°. Yes, because if this is 40, this has to be 80. O RQ is going to be what? 140. I didn't say anything about no 140 here, Liam. Where you seen 140? Somebody get 140? How they get 140? I don't know how they get 140. So, if this is 40, this is 80 double. What will this angle be? Well, these two angles, this is a isoclesles triangle. These two angles, how we know it's isoclesles? This is the center. This is a radius. This is a radius. Same length. So it means these two angles must be the same. Uh maybe 50. 50 and 50 would add up to 100 180. So that's how they got the 50. That's one way to get the 50. Another way to get the 50 is to see that this is a right angle. There's usually two ways to solve the question. This is a right angle triangle. Same semicircle thing. So if this is 40, then this has to be 50 to get to 90 as well. So um this is angles from the same chord in the same segment are equal from the first three ter theorems that's enough to give you an entire circle theorem question. So let's start to cook some questions here. Um undo my stop working. So let's say I have this scenario going on here now. Excuse me. And I was told that this angle, let's give it some names. P Q R S O. Let's put T here as well. And I want to find the uh I was told that QSR is 40°. Now I want you to tell me the value of angle Q R. Find the value of angle Q R and then the value of I think we can find any angle here actually. So QSR is 40. QSR is 40. So what is the value of angle Q? while other people saying 40 Q that's not correct and then how about seeing if you could figure out what the value of OQT is. I don't know if you could see if you can figure it out. So let's do the first one. This is 40. If this is 40, this over here is also 40. Why is in the chat if you understand that? Cuz that's the rule. Angles from the from a common corn in the same segment are equal. And therefore, if this is 40, this is also 18. In fact, I didn't even need to go that far. The angles at the circumference is half the angle at the center from the same corner. Understand that this is coming from the same corner. This is the circumference and this is the center. So, this has to be 80 compared to the 40. That's it. So, how do we figure out what this angle is? Can that even be done? Let's check. So, I can I understand that this is a 90°. If this is 40, this is also 40. Yes, I could figure it out. Maybe, maybe not. I don't know what this angle here is. Do I? If this is 80, this angle here. No, I can't figure it out. I need I need Q. Yeah, it is isosles. We could get the one right here, but we can't get this little slip because I don't know about this angle or this angle. So, I just trying to see if I can figure out that which one you can never use that you can't use it looking like. So, you'll go and see some legitimate questions just now. I know I can figure it out because this angle is unknown. These two angles are unknown. I don't know where T hits exactly. So I know that this is 50 and that is 50. Y'all, I just make up the question. Don't get too deep into it. Yeah. Yeah. Yeah. It's not. So let's keep going. Let's keep going. So the next theorem is the opposite angles in the quadrilateral are equal. Now this is the instruction that I say this 100 times but I'll say it again just in case anybody never hear me say it. When you get the paper please open the paper to question nine. Look for this and seek is spot this. This is one of the most hidden things in the paper. All of a sudden you reach question nine and you forget completely about cyclic quadrilaterals and then boomshot the exam come on Friday. I might do the solutions Friday yourself or Saturday morning I do the solutions and you're like wow it's at a cyclic quadilateral line. know that cyclic quadrilaterals if this is 80 this is 100 opposite angles in a cyclic quadrilateral are supplementary 120 + 60° add up to 180° class y'all with me there opposite angles in a cyclic quadrilateral add up to 180 so we can start to build lots of questions from this then we have a couple simple theorems which are kind of obvious if you split a chord, it makes the radius and it meets at 90°. So that's what this is saying. The perpendicular line from the center of a circle to a cord bisect the corn. I've never seen that term in use like where you have to use it. It's just like so obvious. Two radi gives you an isosles triangle. This is a binary theorem that you use all the time. In fact, we use it for two of the questions already. So this this here is an isosles triangle and the base angles in an isosles triangle are equal. So if this is 45°, this is 45°. If this is 40°, this is 40°. Very very very useful. So anytime you have two radius, it means that this side is equal to that side. So you've created an isosles triangle. Look sharp. You might create a equilateral. Um the angle between our normal and the tangent is 90° or the radius meets the tangent at the normal at a at at normally it will meet the tangent perpendicular at a at 90°. Let me just say at 90° the angle between the um radius and the tangent at the point of intersection is 90°. So the radius itself makes what we call in maths and physics and admats a normal. That word normal literally means meeting something at 90°. Last theorem or second to last term. This one deals with outside. Last theorem. Second to last. Second to last term. Um I get up and walk and take a stretch and take no I am good. I will sit if I have to. The last theorem deals with if you have a point outside of a circle class and it bounces up the circle like this, it actually creates an isosles relationship. So a point the two tangents will make an isosles triangle which is pretty obvious but sometimes this comes like one in every 20 times very rare and if you see it is the most obvious ter theorem to see you're seeing something outside the circle two things come out of one point meeting the circle you're bound to know that this is triangle so if this is 60° this is 60° and usually what happens is that it meets the center like this and they'll ask you the value of this angle if this is 90 cuz this is the normal right so this is the law we just do the radius and the tangent meet at 90 this is 60 this angle here is 30° okay the last ter theorem is the toughest of them to see let me draw it it's it's kind of cropped up here you know my my people playing the fool these days so the last theorems is the one is another one that deals with outside of the circle so if you have a random line that meets the circle like this, that's called a tangent, right? If it cuts through the circle, it's not a tangent. If it meet the circle at one spot, but it still will cut through the circle if you extend it, it's not a tangent. A tangent is a line that touches the circle once and can only touch the circle once, no matter how much you extend it. from that tangent. If I create a random chord from the point of intersection, well, this angle will always be the same as if I made a triangle here that or if the triangle usually triangle is is shown like far up here also. So how it says this is called the alternate segment theorem. The angle between a chord and a tangent is equal to the angle in the triangle in the alternate segment or the opposite segment. So let me see how we write it here. The angle between a tangent and a chord is equal to the angle and the circumference in the alternate segment. So if the chord met here the diagram isn't shown clean here. This is equal to that that I say the tangent fall off. Look down into the next paper. So R just sent me a private circle theorem question. All right. I'll go and check it out just now. Really just now. But let's see if you can get the value of A, B, and C here quickly. Write down what A, B, and C is. Still have bearings to do. From the time you see a tangent and a chord, most likely have alternate segment. A tangent and a chord. Look to see if you're making a triangle. From the time you see a tangent, there's only a few ideas that could come. Right? So a very nice very nice Jonathan plus reason. Now remember for these questions you need to give the answer plus the reason to collect the mark. It's two max. So you need to say what it is calculations and then you need to give the reason. So he gets that this is 26 because it's opposite. See the cycle quadrilateral B. What will B be equal to? 52. Why? Angle at the center is twice the angle at the circumference. Very good. So C, what will C be equal to? If you put it in your own words, you need to cover the important parts. So I just advise just memorizing it and done. Even if I didn't know the the terms, I could memorize the terms, all nine ter theorems now in probably under 20 minutes. All of you all can do that as well, especially since it's a diagram. So, how he gets 64 here? Well, let's see. This is a isoclesles triangle. So, 180 minus 52 is 128. So, these two remaining angles must add up to 128. The base angles in a isocus triangle are equal. You know it's an isosles triangle because it's a radius. So 128 / 2 is 64°. Correct? Remember this is a question that mid-range to top students always understand when the teacher do it. You you're watching me do it. You will always understand. But you need to practice it for yourself. By now you should have practiced opposite. Yes. What is the first thing you're doing in this is literally the first thing you have to do in the paper. You're going to open the paper. Don't forget my instructions and look to see if you see any cyclic quadrilaterals. A foursided figure touches all circum parts points of the circumference. This angle plus that angle must add up to 180°. Here's all the reasoning and the answers there. Do we have another one or no? Let's see if Let's check. Let let me check the private question there. I'm going to check the private question after the close to removing Reero from the like this close. I'm very close. You want to see you're not in the student anymore. You're trying to revise all your notes called represent represent translation in a plane using vectors. I was sure they were in sent me yesterday. Determine represent the location of the image of an object under transformation and the object given the image. So translation vectors. So how much times how much types of transformation we have class? How much sides of transformation we have? Four. We have translation, then reflection, then enlargement, then rotation. Now I I write them in this way because this is in order of complexity. Translation, you just need one vector. reflection. You just need to know the axis or the equation of the line of reflection which is usually like y is equal to x or the reflected in the x axis or the y axis or you reflect it in the line x is equal to one or y is equal to two simple simple lines. Enlargement you need to know the center and the scale factor. You do this over and over. Some of you all probably see me write this down maybe six times. Not even not even joking. center of enlargement and the scale factor. And then the last one, rotation. You need to know the center of rotation, which is usually the origin, and the uh angle, how much you're rotated by, as well as the direction. If it's not 180°, you must put the direction, whether it's anticlockwise or clockwise. Anyhow, so translation is a slide and you use a vector for it. So if you have this point A and I want to translate it let's say I translated 3 4 that means move it across three move it up four or I want to do 3 -4 that is across three and down four well if they give you these two points who can tell me what the what the translation vector is look where a prime is what do you think the translation vector Should I hear the question? We know it's a translation. I want the vector that did the translation. So the translation vector is 35. Very good. Elijah, Ethan, Anah. It went across three and it come down five. One, two, three, four, five. Downward movement mean negative. There it is. Let's keep moving. State the relationship between an object and its image in plane under geometric transformations. So here we see it again from the matrices class if I was you a nice page to do hit them a little memorization x-axis reflection y ais reflection reflection in the line y is equal to x that's this line so like if I have a triangle here it will reflect across there line y is equal to negativex is if you have two negatives on them how this hard to remember all right Elijah let me help You remember this? Everybody locking and let me help Elijah remember this. Elijah, if I reflect in this in the y-axis, right? Watch this. Elijah, watch watch. If this is the point 23 and I want to reflect it in the yaxis, this is the y-axis, right? It's going to come across here. Let's say that's point a. And I'm going to get a prime there. So, if I look at Elijah, what is the coordinates of a prime? What's the coordinates of a prime? No -2 comma positive3. It's still on the tree mark here. What became negative? The two. This is -2. This is positive2. But it's still on the positive three. Right. Vibes. So what happened? The to reflect something in the yaxis. The xcoordinate must lock in. The x coordinate must turn negative. That is why this is negative. A reflection in the y-axis. This is the identity matrix. Multiplying by the identity matrix does nothing. But if I put this negative and I have the point 23 all of a sudden this becomes -23. I have reflected it in the y- axis. It making sense. So when you Yes. You're going across. So when you see the negative on the top that's a reflection in the y- axis. See the negative on the right bottom that's a reflection in the x- axis. What is different of this? Now remember if I multiply by the identity matrix I had a whisp and all because apparently justice for headphone users. If I multiply by the identity matrix the one 01 nothing happens. One 01 nothing happens. identity. That's like adding zero. 4 plus 0. Oh gosh, my camera start to start to be sh class. You okay with that? So, anytime you start doing weird things to the identity matrix, that's where you create your little subtle reflections and your flips and your whatnot. Now, watch this. We switch the one zero. You notice what's different? The one diagonal. Let me show you what will happen there, Elijah. So let's say we have the same point let's say we have the same point here the two three and I want to see what will what will become of the two three 0 by two is 0 1 by 3 is three 1 by two is two 0 by 3 is zero 32 Elijah what you notice this is 2 three this is a prime this is this is a this is the image what you notice it swapped It's swapped. So by doing it back to front all of a sudden 2 three becomes 32 the point reflecting the line y is equal to x. That's how it's do it. And if you make it negative it will reflect in this line. So it'll come -2 -3 that's reflecting the line y=x. So it's very easy to memorize all these things. If you take your time do a little sample take it and just memorize it. Take your five minutes and memorize it. Good and done. Most of you all get in the problem to um learn it because you're not just trying to shortcut everything. So XY um that's the translation. Here's some common rotations. This one flips and changes the um it flips it and it changes the one of the coordinate to negative. So it ends up rotating. So if it's like this, it ends up being a anticlockwise rotation. You could check it out with some points and see. Rotation 180°, excuse me, and rotation 270° clockwise. So, you can memorize this. Enlargement scale factor two. Obviously, if you want a scale factor like three, if you want a reduction, you'll use a fractional scale factor. If you want the type of enlargement that sends it behind the the point and does it like this, then you use a negative number like -3. information coming it here and everything I saying could show up in the exam. So using Pyth what coming up now what is coming here? Anybody have a circle theorem that they want me to explain before I go and do this last 30 minutes of tricks? Am I Nobody. So here's what I'm going to do. Another two minutes breeder. All you run around the house, throw some water on your face. I'm going see if I can finish eat this food here so I can get some energy because what usually happens is that people asking a bunch of question we get end up leaving here 10:00. So 3 minutes break everybody our Mhm. Mhm. [Music] All [Music] right. So, getting ready to look back in. [Music] Yep. All right. So, you ready? Ladies and gentlemen, congratulate yourself. 8 and 1/2 hours have passed. Sorry, my bad. 11 and 1/2 hours have passed and we are now on the last half an hour of time where we are going to be touching um I want to say trigonometry. So let's go. to using Pythagoras serum. Now, if you have some more circle theorems that you want to look at, you can let me know and we'll take a serious questions that no nonsense what this boy sent me here. This boy sent me this Tik Tok meme this this foolishness. Um, so let's go. Pythagoras theorem can be used to find a missing side given the other two sides. Class two things go hand in hand. Two things go hand in hand. Pythagoras theorem and soa they at the same level. So Pythagoras theorem and so toa but they used differently. Pythagoras theorem is if you have two sides and you want a missing side. Class are you with me? You have two sides and you want a missing side. So is when an angle is involved. You usually want the missing angle or the missing side. So you have a side maybe two sides and you want a missing angle or you have one side and an angle and you want the next side. You see it just now. Pyth says that C^² is equal to A square + B square or put in words the square of the hypotenuse is equal to the sum of the square of the other two sides. Lea side excuse me. So we have a few situations where this plays play out really nicely like the 3 4 5 triangle which will show up in your exam was another type of triangle that it will play out really nicely 5 12 13 you know about these triangles cross and there's one with 17 I forget what it was 178 and something who Remember I figure it'll tell me right these are the pythagor and triples but these are the ones popular for C segments no I know eight involved so it doesn't always work out with whole numbers like this for example if I said this is five and maybe this is four four squared cuz I'm and I'm trying to figure out what this is I'll write c a b C^² is equal to A 2 + B 2. So 4 2 + 5 2 C^ 2. So 16 + 25 C². So C is equal to the square root of 16 + 25 whatever that is. 72425 is another one Malik but I'm not talking about that one. 72425. That's another one. And there's there's one with 17 and eight or something like that. I know eight is involved. is 15 178 225 17 is three uh is 15 I could research it so define the trigonometric ratios or acute angles this is soa so anytime you see trigonometric ratios they're talking about soa relate objects in the physical world to geometric properties that use trigonometric ratios to solve problems soa stands for s of theta is equal to the opposite it over the hypotenuse. That's where we get the so to right. What's up, Leila? You pop your hand for kicks. [Music] Yeah, that adm we don't we don't do that in normal maths. That's that's this ESTC moms like adm people head with that adm people your time will come. So use s and cosine rules to use and that's actually a terrible part of ads. Use sign and cosine rule to solve problems involving triangles. The sign rule. We use the sign rule when what? Now this is part of the bearings. We're going to see it in questions just now. We use the sign rule when we have an a an a side and it's corresponding angle and another side and its corresponding angle. Who am I going on another side and it corresponding angle? No, no, no, no, no, no. Use it for any type of triangle. use it usually for a scaling triangle in bearings like real. So, so let's say I have this triangle A B C A is over here, B is over there, C is over there. So, notice how I have angle A and side A, angle B and side B involved in the question. So, two angles, two sides, two pairs of angles and a side. If that's involved and one of them is missing. All right, Lita, I'll look out for you. If you have one of them missing, [Music] um then we can find the answer. For cosine rule, it's a little different. Cosine rule, we only have one angle involved in the question and the three other the three sides. So, one angle is involved and the three sides are involved in the question. two sides hugging an angle and we're trying to figure out this missing side. That's how it's usually used. So like there's a guy here holding up a stick. Some of you all remember when I taught it like that. We know his the length of his two arms. We know the angle and we're trying to figure out the length of the stick. Usually that's how it comes. All right. So let's watch this. Which rule would we use? The sign rule or the cosine rule? Cosine rule for sure. cosine rule is a must for this. So a² is equal to b² + c² - 2 b c cos of capital a yes in the paper but you can memorize it. There's no nothing to memorize it. So 70 ^2 + 50 2 - 2x 70 by 50 cost of uh 110. So we now go into the proof of this eater. We now the proof of it. So 110 you can look it up. It's a nice proof. A squ and if we s if we um simplify this I'll show you the answer. We'll get a big number here and then we have to get the square root. Don't forget to get the square. People forget to do that in the exam for some reason. According to Elijah and Zach going to be 99. So you all understand. No, you're saying something crazy. Square root is when you you find the square root of something. You put square root in your calculator and you put the number underneath. So the square root of 64 is 8. The square root of 225 is 15. The square root of 400 is 20. The square root of 79. We don't know what that is. It's going to be a number between 8 and 9. 8 point something. So you use your calculator to figure it out. The square root of 96 9794.14103 is 98.787 to two decimal places. Look out for the decimal places here. We shall have another question cuz that's the rule. That's the formula. A 2 is equal to B^ 2 + C^ 2 - 2 B C A. Yes. A square here. If I tell you a 2 is equal to 25. What is a? That's five. How I get five? A is equal to the square root of 25 which is five. Relle, you used to know this. I don't understand how you're asking this question. So if I have a^2 is equal to 100 what is a? Obviously it's 10. A is equal to the square root of 100 which is 10. If I have a 2 is equal right 121 yes. Yes sir. So um question two. What do we use here? Is this like another s cosine rule or sine rule? Cosine rule or sign rule? Definitely a sign rule. Why? Two angles involved with the corresponding sides. So if I want to find theta, I'll start with theta on top. The sign of theta over its corresponding side 14 is equal to the s of the next angle over its corresponding sign. Just like that. That's how the sign rule works. The the formula s of a is equal to the s of b over b. So I can work out s of 154. I can put that in my calculator. Raise your hands still up. So the s of theta is going to be s of 15 over 4 * 14. We should have an expression. Oh, and then theta is going to be the sin inverse of whatever that number is. So let's see it in action here. S of 15* 14 is a number, a decimal number. And then theta is the is the state. What going on? if you have to find the value of this sign. What you going to use? All right. You use the Pythagoras theorem. Very good. You'll use the Pythagoras theorem, right? Because it's a right angle triangle. All right. If you have to find the value of this side, what you going to use? can't use so could only use soa if there's a right angle. So that is one p can't use so at all. So what are you going to use? This is the coign. You can't use pythagoras theorem. You could only use pythagoras theorem and so to right angle triangle ladies and gentlemen. So if you don't see no right angle take pythagoras theorem and so and throw it in the bush. That's baby things. You can only use it for right angle triangle. So you need to bring out the big guns. So, it's going to either be the cosine rule or the sign rule because it's not a right angle triangle. Yes. And we shall I told you specifically that a month or two ago. So, you're only using cosine rule and sign rule for non right angle triangles. If you see a right angle triangle, it's better to use Pythagoras as well. Okay. So, Zach, I would make that broad statement and say trigger ratios are only for right angle triangle. The SOA toa stuff, yes, could only work with right angle triangle. In fact, opposite, adjacent, and hypotenuse are words that come from right angle triangles. So, you could only use SOA to right angle triangle. But there are ratios that still appear in this. Look, we still going to use cosine here. This is the cosine rule. Faith, how do we know it's the cosine rule using? Three sides are involved and one angle is involved. Excuse me. Anytime three sides are involved in the question and I only one of them is missing and an angle is involved. I know the angle or sometimes I don't know the angle but I know all three sides. You use the cosine rule. If I have a situation like this now faith faith lock in pay attention. If I have a situation let's say like this and I want to find this side and I know this angle. Let's say I know this angle is 61° and I know this side here is 69° and I know this is 10.7 and I'm trying to figure out that that is the sign rule. How do I know the sign rule is involved here? One angle, another angle. So two angles and a side and the missing side is what we're trying to find. So Elijah, sometimes teachers show you it like that, but it's still sometimes a little confusing when it's like that. It's better to just learn it first principle and then because Yes and no. Sometimes sometimes they might know this side. Like I know this side is 12. I know this is 11.2 and I'm trying to figure out this. So use the words that I am saying. Let me see if I can stop the CD from jumping. Use the words that I'm saying. Kate, if there's a question, it's not a right angle triangle. Two angles are involved in the question with two sides. Two angles. Two sides. What are we using? S if one angle is involved with three sides cosine rule. Simple as that. So let's look at this question from past paper June 2023. I think we did this. Did we do this in the crash course? No, we did not. Maybe not. We did it. Elijah. Anyhow, what is the value of angle A? Anybody here can fill out angle A? Mhm. R you figure out R. R is 78. Good. And if R is 78, what else? What else we know? You all understand how to get the 78 angle at the E will also be 78. That's the alternate segment theorem. So angle at the center is twice the angle from the same um corn at the circumference. So that's what we divide by two. And this you all see the alternate segment theorem. This is a corn. This is a tangent. If this is 78, this had to be 78 or vice versa. So now we know that this is 78. How we going to figure out what a is? It's right here. We can figure out a now. Cyclic quadilateral the first thing that you look at class. So cyclic quadilateral says what? This angle and this angle are supplementary. So let's work out this angle and we'll be able to figure it out. 78 and 48 is how much? 126. So this means this is also 126 here. We do this in questions. So if this is 126, I can figure out the missing angle here which would be something like 64 54 54° I minus it from 180. This is on a Yeah. So E is 78 because it's the alternate segment theorem. The alternate segment theorem says that the angle and a tangent and a chord is equal to the angle in the opposite segment in the triangle. Watch. This is the tangent. This is the chord. This is the opposite side. The alternate segment term is the hardest one to see. So therefore E is also 78. [Music] Mhm. All right. That's the way the cookie crumbles and we'll be able to get that 126. Let's try another one. Oh, let's try a bearings question. The diagram below shows a triangle LMP on the ground. I think we did this one already. Tell you the value of angle MLP. MLP. What is the value of this angle here? How do we do that? Which one we going to use? This is the cosine rule. We use it in reverse. We do it in the um crash course. So, let's just skip past this. Let me just show the solutions. Then they wanted the bearing of L from the bearing of P from L is 210. They gave you this is 210. We just figured out this angle. So, now we'll be able to figure out that bearing. Let's just go through it real fast. These are the reasons. For those of youall who didn't wasn't here for the initial crash course, you could always check through it here the proper way to do it. Um, watch how the cosine rule is using reverse here feet. So you notice how we have three sides involved and we want to find this missing angle. So cosine rule it's not a right angle triangle. We're going to find that angle. So the cosine rule says that a square is equal to b^ 2 + c^ 2 - 2 b c cos a. So think of this as your a this as your common a b c capital b capital c b^ 2 c^ 2 dang dang 150 120 2 y 150 y 120 cos of the angle that I'm looking for angle at a so 180 180° is equal to that and we work through from there you can follow the working I'm just working through calculator bring across these two numbers then divide by this number co of that is equal to that MLP would be the cost inverse and you get 82.8 if this is 82 then the whole thing is 210 we subtract we get this 210 should come like this actually 210 we subtract and we able to get that this little angle here is 127.2 two which is the bearing. Yes. Yes. In this case especially because remember this is the unknown part here. Usually thing that we trying to find is the a like what we do with ratio just now the a is unknown but in this case this was known. So we have to do each part bring across the numbers divide by this little number in front here and then find the cost inverse of the fraction. Practice doing it. That's the same thing as saying ratio. The unknown is in a different spot. The unknown in this in the last one we do ratio. Yeah, but they didn't give you the a square. They give you the a capital A. The angle was missing this time. So, this is what we need to solve for this time. Last time we needed to solve for the sign. Let's keep it pumping. Let's see if I see something new. We already talked about vectors and matrices, right? So, now is the time of 9 minutes. So, I want you all to give me questions and let's just do questions. Anything that's giving you problems, let's just do All right, let's just make up a random bearing question. So, let's say I have a I'll do an exit, too. So, let's do the bearings first. A ship leaves a port Q and heads on a bearing of 270° to a port R. it. Then I put our um 92 kilometers away. It then leaves for a port QRS 27 kilometers on a bearing of Um, let me say let's now come into my 40°. So, everybody draw this and the first question is to draw this information. So, go ahead and draw what you think happened in here. You don't need to send it in the chat, Elijah. You can send it to Fant like my class. Actually, don't send it because remember we have a crash course group. Yeah, you can send it in the crash course group. Send it in the crash course group. Lea all kind of thing. You know, you know profile is a duck. Lea say she think she in a bucket of ice. I just see the duck. Look at the duck. It look like you know in the bucket of ice. Anyhow, just just draw this real fast and send it in the crash as fast as you can. Somebody said it already. No, like from before. Wait, who is this? [Music] Um, who's this? Oh, that's Oh, she sent one to do that one from 2024. What is this? Shreps tonic water. Do not recommend. Everything else was good except that. What is that? What is that? Never drink that before. Thanks for other uses. It tastes good for other uses. You know, other uses that we can't talk about here. So, um, please report Q and head. Everybody have a little diagram. They draw already. Look, this group light up here.