Hey guys here is a quick summary of a whole of your GCSE Maths What I suggest you do is go through your revision guide or free revision guide on my website Make sure you know every single point that any points that you don’t know Go and find you individual videos and check those bits. Make sure you do know them by the time you go into your exam if in the exam they ask you to write It as an integer the answer as an integer they are looking for whole numbers Do not be afraid in the exam to draw out two number lines It is just you [and] the examiner So you are never going to be? Embarrassed or friends are never going to know if you draw a number line in the exam if you need to for a question So if positive and negative numbers especially negative numbers loads of people have problems Ordering these just draw a number line take your time, and fill it in slowly To make your life easier in the exam. I strongly suggest that you get familiar with fractions It is really really going to help you if you know your fraction to decimal Conversions that you are used to putting these in order you can make flashcards you can watch a video that I’ve made doing Flashcards or you can go to our website, save a little time and download those from there But this is definitely something [you] [should] spend time in because of the non calculator paper This is going to come up, and it will make your life a lot easier if you are familiar with this You need to have to use a lot of symbols in gcse maths equals not equal the less than more than Less [than] or equal to more than or equal to If they asked you in the exam for mathematical operation what they mean is divide add subtract or multiply Something and in the exam you’re going to be doing this a lot in your non calculator paper It is going to come up a lot so I strongly suggest that you learn your long division and your long multiplication And for the calculator paper, you know how to use your calculator possibly including bidmas. This is really important When we are adding fractions we need to make the bit on the bottom the same So when I select the denominators same, so I’m going to take you this one here Times it by 2 giving us 2 over 6 Plus 1 over 6 equals 3 over 6 for subtracting fractions again We need to make the number on the bottom the denominator the same so I’m going to times this one now I’m not going time to this one by 3 giving us 3 over 6 minus 1 over 6 Which equals 3 minus 1 2 over 6 we can then divide that by 2 Giving us 1 over 3 as our simplified answer When we’re multiplying fractions we deal with the topic and then we deal with the bottom bit? So we have 1 times 2 over? 5 times 3 which gives us 2 over 15 Which we are dividing fractions the easiest thing to do is just turn one of them upside down So we get 1 over 2 times 6 over 1 Which just like tylium fractions is 1 times 6? Times 2 times 1 which gives us 6 over 2 6 divided by 2 equAls 3 When we are adding decimals it is important that we keep things consistent So you fill it in any spaces with any blanks? And then we are just going [to] go down as normal so 0 plus 8 is 8 3 plus 1 is 4 2 plus 9 is 11, so [we] carry 1 9 plus 5 plus 1 is 15 so we carry 1 and then 1 plus 1 is 2 Taking where the same numbers to divide again. I’m going to put my number in there, and you can see that 0-8 isn’t useful song Lady 13-18 30 minus 18 equals 12 that goes in there 2 minus 9 isn’t helpful So I’m going to borrow a number from here that is going to make that one there 8 that’s going to make about 12 so 12 minus 9 equals 3 [3] years there and then [I’m] going to the 8 minus 5 8 minus 5 equals 3 and then finally one – Theory gives us [one] When we are dividing decimals the first thing we want to do is to get rid of decimals So [we] can move this [in] Place [1] and who is just [almost] [1] in 1 we have to move it in the other? Which gives us? 72 Divided by 4 which is a much nicer form to do so 4 goes into 7 once and we have a remainder of 3 4 goes into 32 8 times 7 Point 2 divided by 0.4 is 18 or [you] can do something similar [with] multiplying decimals So we’re going to shift our decimal place over 1 what we end up with is 3 times 15 3 times 15 is 45 and because we moved our decimal place once in the other direction. We have to move it twice in the other direction Once for moving it this way. It wants to move it this place. It has moved 2 times back giving us an answer of 0.45 at the beginning calculate paper they might ask you to put something into a calculator and just Check you’re doing it correctly and if they don’t let you do it on the connection each paper if it’s in the non calculate paper need to be using bigness or both maths whichever way you prefer with brackets indices or orders division multiplication and subtraction Add a minus operation and divide and times opposite operations So if we have x plus 4 to get x on its own we need to do the opposite we need 2 minus 4 if we [have] x minus 10 and we’re going to give you the most n. We need to do the opposite and add 10 if We have x over 3 and we want to get rid of it we need to do the opposite and times by 3 if we have 4 x and want to get rid of 4 we need to do the opposite and Divide by 4 [I] [remember] our numbers are only divisible by themselves and one and it’s really helpful [just] to Recall the first few prime numbers so the question comes up in the exam. You don’t have to work on out the first Few Are 2 3 5 7 11 17 and 19 when we are looking for factors in these numbers [that] other numbers can be divided by so 20 is divisible by 1 and 20 by 2 and by 10 and 4 and 5 for multiples peugeot’s upwards so 40 and 60 above see multiple of 20 The highest common factor for [number] is a number that is factor of both things and it’s a highest one so you need to look at two numbers you’re comparing work out with [factors] [of] Axford [Lion-o] 1 & 9 3 & 3 facts of 12 are 1 & 12 4 & 3 & 2 & 6 the highest number that is common to [both] of these is 3 The lowest common multiple is something as relational base numbers, but is the lowest common multiple of [base] numbers so if we look at our 8 times table and our 12 times table, we have 8 16 24 and 32 and our 12 times table 12 24 36 and 48 and we can see that the lowest number that is on both of those lists is 24 For prime factorization when you take everything down to our prime numbers remember our prime number of things that can only be divided by themselves and 1 so 2 and 24 We can go into [2] again [Bay’s] 2 and 12 We’re going to again that is 2 & 6 We’re going to 2 again that is 2 & 3 So these are our prime factors that gives us 2 Times 2? [Times] 2 Times 2 Times 3 or 2 to the 4 Times 3 It is well worth learning your square numbers definitely 1 – 13 15 and 20 your cube numbers if you can as well because they’re so much so often it’s Really reinforce that you know is here is my little video. I’ve made on these you can make your own flashcards [I] strongly recommend making flashcards or if you’re on say that I’m going at my cash cards for my website so square numbers So here are your square numbers? 1 squared t squared 3 squared 4 Squared 5 squared 6 squared 7 squared 8 squared 9 squared 10 squared + 4 12 with 13 3 to 15 squared + 20 squared really worth learning those You need as many cube numbers as you can but I definitely recommend the first 5 so 1 q 2 q 3 hughes 4 Cube and 5 cubed and If you know your square numbers and your cube numbers you are going to be able identify your [square] and Cube roots Of many of these as you can remember is really really can help you on the non calculator paper And is massively going to speed things up and if you recognize these numbers as important numbers Then that’s definitely going to give you a hint as to what to do on some of the other more complicated questions It is really important that you can use your calculator properly to calculate pels or 4 or higher and calculate roots square roots cubed roots and Roots the [4] roots of 5. This is really important to just spend some time getting familiar with your calculator Learning which bones to press in which order, so when going to the exam? You know what to do and again police and some time with a calculator Getting to know how to properly use the fractions button to look Keypad in the middle and getting to know how to use a shift key and where Pi is Sometimes long numbers are a little bit of a hassle to write out So we write them in Standard form now what we need to send a form and take our decimal place from there and move it to there So we are going to get to two point seven? Times ten and then we need to look at how many places are different places moved one two three four five six seven So that very long number [there], [so] [we] witness two point seven times ten to the seven works in a similar way for Numbers less than zero we need to move our decimal place from here to here So [I’m] just going to pop it in there We are going to get three point nine, two Times ten and because it’s less than zero is minus and we count jumps one two three four five six seven eight three Point nine two times ten to eight much easier to say and right then this number here if we’re going the other way around [four] point three times ten to the four so three four three and then after some places we need four Digits, so this is our first one here one Two three four we can check this because those [whole] places moved from here to here So [that] should be four jumps one two three four jumps With numbers below zero it works very similar way, so here We have minus six the first thing we can do is write out six zeros one two three four five Six [per] our different place in to our [number] [in] nine to one and you’ll notice [I] different places moved from here to Here so [you] can check the jumps and it should be six one two three four five six? This is another one where it helps you familiar with buttons on your calculator And how they work just in case you have to input this in during your exam It is really going to help you in the exam if you know your fractions decimals Fractions [conversions] is really really worth taking the time to learn these you know what your video is made about this make your own flashcards Or if you’re on Facebook, I’m prepared some website you can download [slash] cards from there As well as using whole numbers in ratios you need to have these fractions in ratios as well On the non calculator paper very likely to get you to use some fractions. So you need to be able to work out What fractions are so 10% of 800 is going to be 80? 15% is going to 80 plus [1/2] so [that] is 120 whereas 200% is going to be double it units conversions are incredibly important and That you need to have to recognize when to convert unit so everything needs to be in the same unit and you need to have To do that confidently so grams into kilograms. There are a thousand grams in a kilogram absolutely this one We need to divide by a thousand and if we are going The other way we need [to] times by a thousand If we are going from seconds into minutes we need to divide by 60 minutes Into hours we need to divide by 60 hours into minutes times by 60 minutes in seconds times by 64 Millimeters in Centimeters Meter divided by 10 Centimeters into meters it is divided by a hundred Meters into kilometers it is divided by a thousand kilometers into meters times by a thousand meters in centimeters times by a hundred centimeters and millimeters at times by [ten] So go from pence into pounds [we] need to divide By a hundred to go from pounds independence we need to times by a hundred Sometimes when we have a number go to a large number of decimals it’s annoying to work with [special] non calculate paper, so we can use estimation so [3.94] [1/2] can be rounded up to make four and three point zero zero zero one can be rounded down to make three Which is a much easier some fast to do you might be asked whether your answer is an adverse station on Underestimation if your rounded up is going to be an overestimation brand down under [estimation] if it’s in a fraction It’s slightly different. So if [you’ve] rounded up, and you made it a denominator it is then going to be an underestimation Rounding is an important skill. So here are two numbers We will go around to two and three significant figures so [the] first thing you need to do is to look for your decimal places Where your your significance [Bieber’s] going to be so choose [consistently] as we want these two? We then need to look at this number here because it is above five to two significant figures This is going to be nine four and then we need to replace the rest of numbers with zero so nine come [0t] From 0 1 from 0 through from zero to become zero and one [from] zero for three significant figures these are three numbers [we’re] going to write down So 9 3 9 We then need to look at additional, but is below 5 so we don’t need to round it but we do need to write them in in place as zeros so 2 1 3 2 1 the same for decimal places We are going to go in first two decimal places so that is these ones looking at this number here. It is below 5 so it’s going to be nine point three nine And because it’s below [zero] we get out here to pay attention to the rest of them Looking at three different places it is these ones looking at this number here. [we’re] going to get to nine Point three nine [to] Here we have an inequality and our number x falls between these two numbers So it is greater than or equal to minus five and we send all that on a number line by putting in a field in that circle or Less than mine [or] less than [four] so that is an open circle and our inequality will fall in this region here One way of looking at the limits of accuracy We’re talking about combining rounding and in this case a case inequalities So what is the lowest possible value that could round to minus 5? So if we have a think about this – five point one so once in this configure is going to be minus five So is minus five points 3 and minus five point four but? Minus five point five is going to round [to] minus 6 so our lowest possible value is minus five point five our Highest possible value that is round two less them for but not equal to four and this is something That’s a little bit [complicated] It’s going to be three point four because if it was three point five that would round to fall But our number is not equal to four it is less than [four]. So these are our limits of accuracy if you’re doing the foundation paper you have now finished and this video you can go and watch the [other] videos go and try some of the example walkthroughs to check that your knowledge and That you can apply what you learnt or if it in high t paper, we are a few more bits to go You need to have to estimate roots for any given number And this is another instance where learning your square and your cube numbers comes in will be useful so if we are trying to find the square roots of [39:39] Falls in between two square numbers 36 and 49 and 39 Falls pretty close to 36 so you can see that [will] start with a six and is going to be quite close to 36 put one seems a bit too small so six point two is a rough estimate Look for the square root of 39 Another thing is well worth spending time to calculator if you learn how to input served properly so that you don’t make a mistake when you are calculating with And you also need [to] know how to simplify expressions using sides So here we have square root of [5] squared bass games be the square root of 5 squared Square [root] of 5 squared is Simply 5 – square root 5 plus 3 Square root 5 is just 2 plus 3 added together so it’s going to be 5 square root 5 – square Root 5 Times 3 Square Root 5 This is getting slightly more commentated now all we need to do is put everything inside Square root so 2 squared is 4 times 5 times 3 Squared that 9 times 5 what we end up with then is 4 times 5? 29 Times 5 45 20 Times 45 900 and then we can look at that see if we can divide that out into any square numbers And we can that is 100 times 9 so square root of 100 is 10 Square root of 9 is 3 So it’s 10 times 3 which [equals] 30 as well [as] the [non-return] decimals that come up in the foundation – you also need to convert to between and decimals and fractions will Reoccurring ones as well [it] is so we’re spending on time just [learning] these it will make your life so much easier in it the exam If you have three things to do in order [and] need to know about the number of ways of doing these different things So task one say there are a number ways of doing it Task 2 So there are being number ways of doing it And [I] [squeezed] either a see number of ways of doing it. You could make a very very long list and work it out, or you can do a Times B times C equals the total product for this example Fractional indices look scary and perfect for actually bad not they’re just opposites so 8 squared is 64 and One over two as our indices is exactly saying [the] same square root So one over means like doing the opposite So 64 to power of 1/2 is square [root] of 64 which is going to be 8 so? 3 cubed is 27 and 27 to a third, so it’s the opposite of Doing the cube, so the cubed root of it is going to be 3 if we’re combining things we just follow the laws of indices so 9 to 1/2 times and 1/2 square of 9 times squared 9 because [its] indices means we add them neither one that just gives us 9 if we have a Vacuum or communicated indices and fractional indices then we need to break it [down] into path, so this is basically saying the Square Roots of 4 cubed so the Square root of 4 is 2 2 cubed is 8 Sometimes in math writing things out fully can be really really tedious. So there are a few conventions that we use to speed things up you have to [be] able to [rationalize] what they mean and you have to be able to use them in Algebra, so [x] times y can be written as x y we don’t need to bother writing the times m if we have x and x and X again, that’s writing out quite a lot of things we just have three x’s x Times x with [y] is x squared and then if we want to have x divided by y we can try it out of x Either y in EDL to put terms into an algebraic expression so here we have 2 x s 3 and x equals 4 so what we need to do is replace the x with 4 [that] is 2 Times 4 Plus 3 equals 8 Plus 3 gives us 11 [n] is how to collect like terms in an algebraic expression and for this what I stated is is we write things in order or Get your highlighters out and start highlighting So we have x’s things you like seeing things [as] x in? This is why in things is yn. Things with y in now we can rewrite this as x plus 2 x plus 3 [x] all together that makes 6 x Plus 1 2 3 wives I Can multiply out a number in front of our bracket so here is our 2 in front of our bracket we need to turn it? In by everything inside the bracket to get rid of the brackets So we have 2 times x gives us? 2 x Plus 2 Times 4 gives us 8 Whenever you see a long complicated algebraic expressions It sounds quite simple, but you should always consider factorizing it to factorize it anything for what both things common So we both have three white Amber 12 can be divided by three, so they’re going to put that outside We’re going to make some brackets three [y] divided by 3 is y 12 divided by 3 is 4? You must go out [to] expressions now This is a things that get a bit complicated And I think it’s really important that you draw your lines in even if you are that really good this please just draw your lines in because in the exam you might get a bit stressed you might make a Silly mistake that you wouldn’t otherwise makes please draw your lines in there were two different ways of drawing your lines in You can go one two one two All the way that I prefer to do it just because it looks a bit a little bit nice stuff [is] like this and then we can turn it into a smiley face now I know. This is really really trivial, but exams are stressful situations, so Anything we can do to make it a little bit Nicer a little bit happier a little bit less stressful Is going to be a good thing? So there is my expression? I need to do this in whatever color you like we still have some multiply it out The other thing I would like you to do is once you have finished doing the line just cross it out So that you know you’ve done it. You don’t repeat it So first of all I’m going to do this line here that gives us x squared And I’m going to cross out the lines. I know that. I’ve done it, but I’m going [to] do this line here three times four then I’m going to cross out the line, so I know that I’ve done [it] then I’m going to do this one three times x Then I will cross out the lines. I know that. I’ve done it then I’m going to do this one x minus four Then I’m going to cross out the line then because I’ve done that one then you need to collect the terms together So these two these two Those two are the same so [you’re] going to need to put them together and you need to sort things into the right order so x squared plus 7x Plus 12 now Two different methods up here this one will give you things in the right order Whereas the Smiley face method won’t give you things in the right order usually remember to put them in the right order That’s writing things that when [former] x squared plus bx plus C. So for this one here. We need two numbers [that] multiply and For this number here, we need two numbers [that] add So first thing you do is you write down your two open brackets? Since this is just x squared we can pop that in already x since both of these are plus We can pop that in as well Then we need to think of all the things that multiply to make eight so we can have a one and eight We can have two and four I Think that is everything nailing to look at your list of numbers Can we use one and eight in any way to automate six no? We can’t can we use two and four in any way at all to make six. Yes, we can two and four Sometimes your algebraic terms will begin and quite complicated. So you want to make things as [simple] as possible I once imply an algebraic expression you can need to collect like terms You [need] to factorize you might need to deal with some Black is expanding brackets in there you also might have to deal with some fractions awesome thirds Rearranging equations to make a different thing subjects the formula comes up So frequently in so many different places is definitely one of the core skills. I recommend you practice a lot so here We are going to make [y] the subject of the formula so first thing I’m going to do is sum minus 6 on both sides So we’re going to get 8x Plus 4 minus 6 Plus 4 minus 6 is going to give us 8 8 x minus 2 equals 2y you can see why it’s coming times y 2 so I’m going [to] divide that by 2 giving us 8 x -2 divided by 2 Equals 2y divided by 2 you see here. I’ve [got] [two] on top and bottom so [it’s] going to cancel here I’ve got numbers I can divide so that is going to give me four [x] minus 1 equals Y it does not matter that y is on this side the equation Is exactly the same happening on this side it just happens to be on the right-hand side do not feel you have [to] Minus 2y at any point to get y over here and then y x get x over this side is perfectly okay to have y on at the right hand side, so another example where knowing your square numbers and your square roots This is going to come in really really handy answers are not scary They’re just a nice easy simple way of writing a long horrible number For example square root 6 is much easier and more accurate than watching the number out in full But you can just treat them like anything else in Algebra 2 square root 6 plus 2 square root 6 all we need to do is to add the numbers in Form just like adding 2x and 2x and we are going to get 4 Square Root 6 Again, just like Algebra if you have 6 square root 6 divided by 2 3 6 6 divided by 2 is going to give us 3 Square Root 6 if We have square root of 2 times group 18 that is exactly the same as saying square root of 2 times 18 2 times 18 is 36 the Square root of 36 is 6 so [now] they look awful, but they’re really really not I promise [if] We are multiplying terms. We need to add the indices for example x [cube] times x – 7 we need to add the indices 3 times 7 or give us x to the 10 if We are dividing terms. You need to subtract the indices for example x is 6 divided by x – 2 6 minus 2 is 4 so it’s going to give us x to the 4 functions another thing that looks awfully scary But really [aren’t] so Fx equals 3x plus 2 when used answer if s is 3 so all you need to do your notice here [excellent] [replaced] by three, so in here we need to replace x with 3 So that it’s going to give us 3 times 3 Plus 2 3 times 3 is 9 plus 2 equals 11? Coordinates are going to be in the format x and then y So x and then y and vectors not relevant here, but kind of so excellent Op [and] white on the bottom so if you’re plotting points We need to go x first so 2 across and then 4 up So our first coordinate will be there And we mark [that] with a nice cross with a sharp pencil [a] blunt pencil is not going to be any good in the exam For our second set of coordinates. We go minus 1 minus 3 down there again sharp pencil if You’re plotting a line from an expression see y equals 2x plus 1 the simplest thing [you] can do is work out some points So if y equals 1 so when x equals 1 y equals 2x plus 1 so when x equals 1 y equals 2 times 1 which is 2 plus 1 making of that 3? When x equals 2 we have 2 times 4 2 times 2 which [is] 4 plus 1 giving us 5x? equals 3 it is 2 times 3 6 plus 1 7 we can now plot those points on our grid and using a Ruler which I [don’t] have You can join the prints up When we’re trying to find the gradient or the insect of a straight line and we need to be thinking y equals Mx? plus c M being the Gradient and C being the intercept So if we look at finding the gradient first what you need to do with a pencil and a ruler is draw the biggest triangle that you can [fit] on bigger triangles are better because they aren’t more accurate And so I’m going to get in there because it’s a nice number, so we are going from 5 to minus 3 that gives us 8 Then we [are] going from 3 to minus 1 that gives us 4 so to find the gradient it is up divided by a cross so up, we Went 8 across we went for So 8 divided by 4 gives us a gradient of 2 And then C is just where it crosses the y-Axis? It crosses the y-Axis at minus 1 so our C. Is minus 1 giving us a total equation of Y equals m which is 2x C which is Minus 1? if Two lines inform that way it was an [extra] [see] are parallel they don’t have same gradients, or the same m So it doesn’t matter where they cross the y line if they have same gradient? They will be parallel? for a quadratic class The roots are going to equal the x intercept And the [x] [intercept] are going to these points here on the [x] [axis] Turning point is going to be the coordinates of this point here The graph of y equals x is going straight [through] the middle Increasing a number in front of x is going to make it steeper Decreasing [Membrane] [font] x is going to make it shallower Putting a number after it is going to shift it either up or down You need to recognize the shape of the graph of x squared which is nicely like a big smiley face Unfortunately minus x squared is less a third race pinning a larger number in front of the x squared say 2x squared squashes it and a smaller number makes it expand outwards X plus a value squared is going to shift it left by that value X squared plus three is going to shift it up by [that] value Here is the graph of y equals x cubed? [minus] x cubed 3x cubed with a large number in front of the x cubed 1/3 x cubed with a small number in front of the x cubed x cubed plus a number will shift it up or down by that number and X plus number and then keys will shift it left or right by that number here is a graph for 1 over x The important notes Remember when you’re plotting graph is that your points are accurate a sharp pencil And a cross especially to accurately plot points that they are clear that you do not do a massive circle Our massive splodge or your pencils, not sharp enough So there’s no exactly clear to the examiner where you’ve putted it on one box or another box your axis has had a clear Scale on it The scale has to be an order scale has to make sense it has to have a title and it has to have a unit And your line needs to be a smooth curved line this sort of line here Is not going to get you any marks in the exam remember this is maths not arts. We need a smooth strong confident line and a line [of] best fit distance time and velocity time graphs look identical The only difference is that one is labeled distance and one is labeled velocity or speed And you have to be paying attention to the graph Before you can answer the question [because] either [mosi] looks the same as having you very very different things so if we first look at this section here on A distance time graph this is going at A study speed the Middle Section here on a distance time graph You will see that as [time] is progressing distance is not progressing. So this is staying still Then the last section they are also moving at steady speed but if we look at the slope or the Gradient of the line you will see that this steady speed is slower than this steady speed here if we look at a velocity time graph the first part is acceleration The middle part where it’s flat [so] time is going across a velocity is not increasing But velocity is not zero so they are still moving so this is now a steady speed and Then the last part is again acceleration But if we again look at a slopes, you’ll see the slope here is shallower the gradient is shallower than this one So it is accelerating less Rapidly if we want to solve to see linear simultaneous Equations from a graph we need to be drawing the points on the graph We need to be looking at where they cross over and that is going to be this solution if you have a linear and a quadratic equation Again, you need to draw them on the graph and the points where of course over this is going to give you two solutions is the answer Here we have a quadratic equation that we need to solve by factoring so we need to draw our brackets x X you can see we have a plus and a minus so they’re not both parties two can’t like that in there the moment we need to look at numbers that multiply to make four and then add or subtract to make three so we can have one and Four or we can have two and two there’s a bit more complicated in that because it’s minus four So we can have minus 1 and plus 4 or we can add plus 1 and minus 4? minus 2 and plus 2 but they need to add in some way together to make [3] and There’s nothing we can do with 2 [&] 2 so you can’t leave that It has to be plus 3 so minus 4 plus 1 is going to give us [minus] [Teresa’s] not that one so there has to be this one here, so we are going to have plus 4 and minus 1 applause my brackets and Once 4 Plus 4 and minus 1 now we know this equals 0 now something to equal 0 two things times by each other equal 0 one of these has to equal 0 So to make each bracket equals 0 x plus 4 equals 0 we need to rearrange that [so] x is minus 4 or x minus 1 Equals 0 Which rearrange will give us x equals 1. So there are two solutions to a quadratic? Equation by Factoring you have to put it into brackets and then make the brackets equal 0 Simultaneous equations can be solved in a large number of different ways you can find the way that Works the best which you prefer and then go with that one Substitution which is for no particular reason one third method? multiplication addition subtraction or you can [fill] [them] graphically graphically is it more [complicated] and and Leslie asked you to do it Don’t generally do it that way So I’m going to show you two ways of solving this same equation [and] hopefully we should come up with same answer both different ways So I’m gonna call this equation number one. I’m going to call this equation number two, and I will start with substitution So I’m going to rearrange equation number one to make wireless objects the formula, so 2x plus y equals Seven y equals 7 minus 2x So now I know what y is I can pop it into equation number two so 3x minus y equals 8 3x minus y which we know from previously is 7 minus 2x? equals 8 most kind of brackets 3x minus 7 plus 2x equals 8 tie this up to 3x Plus 2x gives us 5x 8 plus 7 gives us 15 15 divided by 5 x equals 3 Now we can now we know what x is. We can put that into either equation number [one] or a crazy number, two I’m going to equation number one 2x plus y equals 7 x equals 3 So 6 plus y equals 7 and Rearrange that so y equals 1 so that’s doing it by substitution [I’m] now going to do it by addition now addition subtraction and multiplication all go together because if you want to have Addition you need to have something with the same number in front here I’ve got one a [100] one but different symbols if I add these together they’re going to cancel each other out Subtraction [would] work if this is plus y and plus y and if we didn’t have any of the numbers same here? We could multiply one of these equations and for example. We could multiply this one by [1/2] if you wanted to get X the same so we could either add or subtract those so those three work together, so I’m going to add the equations together so 2x plus y equals 7 3x minus y equals 8 Add them together gives me 5x minus y plus why gives me zero 5x equals 15? Divide that out x equals three now that x equals [three] I can pick either of the equations Since I picked equation [um] one last time I’m going to pick quiz number [t] this time so 3x minus y equals 8 x equals 3 so that’s 9 minus y? equals 8 Take 9 over slightly at minus y equals minus 1 Times at 1 minus 1 y equals 1 so you can see both methods give the correct answer both methods are Valid you just need to pick which method you prefer the next part making equations from a situation given in text can be tricky [for] some and when I’m working [use] for exam papers I try and Do this for you as much as I can pilot because math makes more sense in my head than words does and Partly because you need to know how to do it for your exam So primrose went to the shop and she bought 4 biscuits and one [sweets] this cost her 124 the next day she went to shop again amble to fisk and [two] sweeps this time it cost her 199 how much is you sweating biscuit cost now on the face of it this may look like a hard it made it look impossible Question [4] what I’ve done is I’ve taken information texts I’ve written it down as two equations And I can solve it as a simple simultaneous equations question don’t try and [solve] this I literally just made it up [so] I don’t know where there are actual answers to this, so please Don’t leave those comments about how I’ve done the equation wrong Several inequalities can scare [people] because of this big in the quality here in the middle But you can just treat it exactly the same way as you can and equals so air cross 3 is greater than seventh meeting the [300d] outside – [and] [three] gives us x is greater than 4 now we need to pop that on a number line So we can draw an open locals at four Because open circle shows that it’s greater than if it was greater than or equal to if it had This symbol we [would] do filled in circle But it’s just greater than and then we are going to go this way now There are a few things you need to be aware [or] [volure] solving inequalities The sign does not change direction if you add or subtract numbers or if you divide by a positive value But sign will swap so from greater than to less than if you divided or multiplied by [a] negative value If you are given an empty sequence a n plus 4 you need to outline any sequence in this term So when n is 6 n plus 4 is going to equal 10 and when n [is] 100 it is going to equal 104 now slightly more complicated that is doing it reversed finding the NTH term from a sequence the first thing you need to do is to work out the differences, so [between] 1 [&] 5 is Plus 4 between 5 & 9 it Is Plus 4 between 9 and 13 [it]? Is Plus 4 telling us the first part is going to be 4 n? Now we can actually work out. What 4 n is So if n would equal 1 4 N would equal 4 if n was 2 for second term [4] n would be 8 if n was 3 For N It would be 12 if n was for 4 n would be 16 Now we can work out the difference between what the term would [be] if it was [4] n This is for [N]. And then we’re going to work out [the] difference so 4 minus 1 is 3 and 8 minus 5 is 3 12 minus 9 is 3 16 minus [13] is the ring and that gives us the second part which is minus 3 for N minus 3 Your square numbers cube numbers and high numbers are going to come up a lot so is worth recognizing them Least but preferably what I would like you to do is try [and] learn as many as you can For square numbers that is 1 2 13? 15 and 20 So 1 2 13 15 and 20 they will speed things up if you keep pressing those those as many Cubed numbers as you can and some of these triangular numbers as you can you can see over here that I’ve made you a video going through all of these you can flash cards are a great [thing] to do or even though this website and Download the ready-made set of flashcards to speed things up Foundation to students well done you have finished this section And you can go on to do another section or go on and test yourself with some past papers or go through the revision guide And work out which bits you need to work on which bits you okay on high Oc students I’m afraid we still have quite a lot left to do Third in fractions in things sometimes freaks people out. There’s no reason. Why it should do that You can either work with the fractions as they are and simplify it or you can get rid [of] fractions And [you] just need to get rid [of] fractions here is two times everything by [4] so 1/2 times 4 is going to give us 2x 3/4 times 4 is going to give a story And then 1/4 times 4 is going to give us 1 now isn’t easy one to solve you could just leave them as [they] are And [filled] it like that So we are going to have 1/2 x equals 1/4 most 3/4 That’s minus 1/2 so x is going to equal minus 1 let’s just check the other side moving 3 over We’re going to get 2x equals [minus] 2 so x equals minus 1 Slightly more complicated quadratic to Factorize now still them starting the same weight by drawing down our double brackets 2x has to go in one of these because the only thing that times is to make 2 is 1 and 2 so 1 and 1/2 1 1 I’m happy to if it was 4x It was slightly more complicated because it can be 2x and 2x or it could be 4 x + 1 x now remember? We need to have things [multiply] to make this and Add to make this one so things that most try to make [6] Are 3 & 2 1 & 6 But your notice is minus 6 so that is a bit more complicated so it could be minus 3 and plus 2 or it could be plus 3 and minus 2 So minus 1 and plus 6 or plus 1 and minus 6 but because we also have to be 2 in front there We don’t know which way round they [go], and this time it does matter So it could be minus 2 and plus [3] or it could be plus 2 and minus 3 or it could be minus 6 and plus 1 or it could be plus 6 and Minus 1 so we actually have quite a lot of combinations to go through here now We need it somehow in combination with [2] to add to Make 6 and then looking at this initially if it wasn’t a Certainly more complicated quadratic you would initially assume. It’s this one because these ones easily combined to Give [+1] but we have taken into account that one of these numbers is going to be multiplied by 2 So you need to little bit trial and error you need to do a little bit thinking and without thinking Even if one of these numbers was x by 2 there was no way, we can make it add to make one So we can discount all of those problems. Who’s at the beginning? Now what you need to do is to put a few things in and try and work it out now We know one of these has [to] be x by 2 However if We x is 3 by 2 that’s going to give us 6 and there’s nothing we can do to 6 [&] 2 to make it equal Plus 1 If we times a 2 by 2 that gives us 4 4 & 3 we can combine to give us plus 1 so? From that we can assume that it has to be the two That is x by 2 and the 3 that isn’t x by 2 now? we just need to work out where our pluses and our – is go and We need to come up with [+1] so they don’t have had plus two times by two giving us plus four minus three and These can seem quite complicated if you go through and do Every single possibility by trial and error that will take quite a long time These you just sit think and try and apply a bit of logic before you start with trial and error that should narrow down the number of things that you have to try a directories can seem complicated but in adjust and the application of logic and Algebra together there are a few short hands that you need to know to make your life easier if You need to have an even number It is 2n Because even if n is odd number say three times by two six that is an even number If you want something [that] is an odd number You can do to N minus one with an even number Minus [one] or plus 1 is going to be an odd number if you have consecutive numbers you can have n N plus 1 and N plus 2 and if you need to produce something is a multiple of something else what you need to be thinking about is Factorizing, so if you need to create something is multiple of six then you can take Whatever it is and as long as you can divide six out of [it], then you proved is a multiple of six Inverse functions may seem complicated, but they’re not you need to do take your function And then think about what you’ve done do it backwards and do the opposite so with [three] x plus [two] you would times by [three] and then add two if we want to find the inverse function of this You would need to do it backwards So [you] would need to do the opposite but going backwards So [instead] of adding [two] last we would – two First and then instead of x in by three first we would divide by three last So what we are going to get is x minus [two] over [three]? Constant functions are combining two functions at once and what you need to do is look at the one that [is] Closest so we need to do g to x first And then once we’ve got our answer we need to put it back into f So if we put g if we put x into g we are going to come up with [two] [x] plus one so then we need to put that into f in place of x So we’re going to get three and so writing x I’m gonna write my answer from G. Which is two x plus one Plus two now [we] need to multiply out the brackets So [three] x 3 times 2x gives us 6x plus [3] plus 2 so 6x Plus 5 if a and B are [perpendicular] lines on the graph then the Gradient of A And the Gradient of B together are going to equal minus 1 for example if a gradient of a was 1/2 and We knew that [if] [they] [were] perpendicular? They would have to equal minus 1 so We’re trying to find the gradient of the what we would need to do is to rearrange the equation see B. Equals minus 1 Over 1/2, so it’s going to give us minus 2 the turning points on a graph is the [bits] down here where the Gradient changes now you need to rearrange your quadratic equation So it looks like this and as a formula Then we can say the x plus a [squared] minus b Equals 0 and note even I’ve used a and a [and] B and b here these do not necessarily mean these are the same values and once you’ve got that you can just rearrange it to find what x going to be and Once you have x you can then find y The graph of y [equal] to 2x will look like this The graph [of] y, it was half to the x would look like this both of these passing at one You need to recognize the sine Cos and tan graph so your [stine] graph is going to start at 0 it’s going to get up peak at 90 Back through 0 at 180 down 2-1 at 270 again through 0 at 360 peaking at 1 at 450 and then back down to 0 at [Sli’s] 40 your [cost] graph looks very similar it is shifted over slightly So at 0 we’re going to be at 1 and then going through 9 to go to 0 at 90 down to minus 1 at 180 back up to 0 at 271 it’s 360 0 at 450 and then minus 1 at 540 your tan graph does look quite differently so starting at 1 we go up And then we go just past 90 ever so slightly just past 90 we’re back down again 0 at 180 and then these are going to touch either side of the 270 and [450] Lines This is just combining things that we know about functions and things that we know about graph So instead of just looking at the function as Algebra if we sketch it graphically you should be able to sketch both of these graphs [x-Squared] + Gx equals 4x minus 5 and then think about the things that you’re used to doing to graph So making them wider make them narrower for quadratic graphs moving them up down left or right? So this isn’t anything new here. This is just combining, two slightly different things to in one question Sometimes in maths they need you to be very accurate and specific and sometimes they will need to estimate an answer So if you want to work out the gradient of graph because it’s telling us something like in this instance the gradient of a velocity Time graph is the acceleration if you want it to be Exact and we have a straight line. We can look at a change in velocity over change in time however here We don’t have a straight line. We have a curved line. So if we just want [an] estimate Say over the first minutes We can just look at the points where it ended up where it started and used those points Assuming it is a straight line to then work out the change of velocity changing times work out the acceleration This will not give you an exact answer such questions as calculate this not [what] we do this is what you do is the question is asking for an average or if the question is asking for an estimate to [work] out the distance [travelled] from a velocity time graph We need to work out the area under graph now the best way to do this is to divide this up into sections Like this this section a b C and D Now oops another section there sneaky look for that being – and Every time you see a change in velocity or changing the graph put in another section Now to work out the area of a [is], just the area of a triangle [areas] B. Is the area of a rectangle? b – [you’lL] notice we have a Rectangle here with a triangle sitting on top of it every C Is just a rectangle whereas with D again? We have a Rectangle with the triangle sitting on top of it then add those together and you have the distance [travelled] From a distance time graph working out the distance traveled is just going to be reading off axis here Working out the speed is going to be distance divided by the time so it might just ask you for section Or it might ask you for the whole work thing From a velocity time graph if you want to work out a different phase area under the graph The speed you are just reading off here the only difference between speed and velocity is the velocity has a direction when it’s combined quite a lot to find the Creator [of] tangent from the equation circle first thing that we know is [that] the tangent is perpendicular radius? So if you’re trying to braiding at the radius by the Gradient of tangent, We should get minus 1 To find the gradient of the radius you [can] just turn it into a right angle triangle [workout] up which is the wide point once [workcenter] divided by the across Exponent [takes] Center that give you the Gradient Pop it back into this equation Make it [equals] minus 1 and from that she’s worked out the gradient of the tangent you can then use the equation wyatt – and The work will [network] we have from our point. It was a gradient Which was just worked out here times x minus the [x-coordinate] that we have from our point to find the equation of a tangent? If we’re [solving] for exact equations by completing the square We are forcing our quadratic equation into something [that] looks like this now if we [think] back to multiplying out brackets this number here times this number here is going to come up with our x so that is going to tell us what this value here has to be so for this we only need one brackets and If we are going to get something plus something which is Gonna be the same thing equaling 6 x it has to be x + 3 squared inside the brackets now what we can do is multiply out [x] + 3 so it’s going to give us x squared + 6 x + 9 You can see we don’t actually want [plus] 9 here We want plus 7 so to get from Plus 9 plus 7 we – 2 so our answer is going to be x + 3 squared – – If you’re going to solve an equation using iteration this is just slightly more [sophisticated] method than try an error So you’ll generally give an equation You generally given [and] [a] clue as to [webstart] or you’re told that your solution lies between two values and you need to be Vertical you need to be inaudible. You need to explain what you’re doing at any examiner’s otherwise. I’ll get lost So start with the values that they give you Work out and what the answer is going to be in here and then write whether it is too big Too small with your close, and then maybe go up in [licks] bits You might have to make a big jump you might have to make a teeny teeny Tiny jump But it’s all about making small jumps commenting little small jumps letting the example know your logic So that [you] can work out what the answer the closest answer you can get is going to be? solving linear inequalities using a graph is very similar to solving Equations using graph [in] to the sexual point you’re going to have a whole area that fulfills your results So here is my inequality and you need to work out which area of the graph is going to fill your result What you might get is a second line on there as well, and you’ll be looking [for] the areas that Overlap so say this area and this area [so] this area here would still the answer for both? inequalities the last bit here for High-tech Algebra quadratic and sequences so very similar to our Previous sequences first thing we need to do somewhere count difference there we had [1344] Minus 25 is 1969 minus 44 is 25 100 minus 69 is 31. They are not the same like [you] would expect in a non quadratic and [term] so we need to work out the differences again 19 minus [10] is 625 minus 19 is 6 31 minus 25 is 6 so we’ve had to go down to layers that tells us is N squared The difference between these is 6 because N Squared [we] [need] [to] divide that by 2 so that [gives] us the we and squared? Now we need to work out what 3 and where’d is if n was 1? 2 3 4 5 [I] said 1 Squared times 3 is 3 2 squared is 4 times 3 is 12 3 squared is 9 times 3? gives us 27 4 3 16 times 3 gives us 48 5 squared is 25 times 3 gives us 75 and Then we need to work out what the original value minus 3 and is and we shift that up. There is give it so original minus 3n squared Gives us so 12 2 3 3 is 9 25 to a 12 13 44 Takeaway 27 is 1769 minus 44 is 21 and 100 minus 75 is 25 now [we] can do the same work out the differences Between them, so 13 minus 9 is 4 17 minus 13 is 421 were 17 is [425] most 21 is 4 so that will give us plus 4n Now we need to work out what? 3n squared plus 4n is which will give us values of 720 39 64 and 95 and what we need to do is original? minus [3m] squared [plus] [4n] and for every single one of those three a difference of 5 so this tells us our total term is 3x squared plus 4n plus 5 to get from seconds to minutes [there] are 60 seconds in a minute So you need to divide that by 60 to get from minutes into hours there are 60 minutes in an hour? So you need to divide that by 60 to go the other way to go from hours into minutes? You need times by 60 and then minutes into seconds times by 60 Grams into kilograms you need to divide by a thousand kilograms into tons divided by a thousand tons into kilograms times by a thousand kilograms into grams times by a thousand now [meters] second into [kilometers] [per] hour is least bit tricky. We need to do several steps the first step is times by 60 The second step is times by 60 and then the final step divided by a thousand and then in reverse And we want to be from kilometers per hour to meters per second it is times by thousand divided by 60 divided by 60 and if you want to go from pence into pounds there are a hundred pence in the pound so we need to divide By a hundred to go the other way we need [to] times [500] and it is the same for prices One gram per centimeter cubed is equal to a thousand kilograms per Meters Cubed So slightly different here when inter times at five thousand and then going the other way we need to Divide by a thousand there are barrels and pascal’s in kilopascal So to get from Pascal’s to Kilo Pascal’s and meters help the other one was unusual [needed] [divided] by thousand and this one we need two Times five thousand To go from centimeters into meters there are hundred centimeters in a meter So we need to divide that by a hundred again other way Winning two times by a hundred now as we gave two from length two areas [of] volume it gets a bit more complicated because in one Methyl we have a hundred centimeters one meter a hundred centimeters one meter 100 centimeters, So one Meter squared is not equal or is not the same I Was thinking about though from centimeters to meters because we have a hundred times 100 which is 10,000 centimeters squared so to go from centimeter squared into meters squared we need to divide by 10,000 meter squared in centimeter squared is x by 10,000 we can think about Cubed in the same way because one meter cube is one meter by one meter by one meter so one meter is 100 centimeters one meter 100 centimeters one meter 100 centimeters, so one Meter cubed is going to have One two three four five six zeros after it one two three four five six Centimetres cubed after it, so this is a bit more complicated going from centimeters cubed into meters cubed We need to divide by 10 Liam three four five six So I’m going the other way times two three four five six To be able to use scale factors scale diagrams and maps you’re going to need to get your ruler out So please remember to take it into the exam with you and you’re going to need to I measure really actually I’m talking millimeter measuring here and For every for example one thing to me to measure on the map that could be three point seven set kilometers in real life so you just take whatever you can measure on the map on the Exam paper [and] times that by the scale Factor given to you in the exam Ratios can come up in a lot of different ways one of the ways in come up is in particularly Worthy questions where it doesn’t necessarily look like a huge question But you have to work out and you have to take the rule of context and then spreche the relationship as a ratio, so a party there are men and There are women there [are] eight men for every six women if there are? 24 women how many men are there now you’ll notice what ever since out? I’ve left a big gap in the [middle] here that is because I’m going to put one in here and This is how I’m going to work out my ratio So what you’ll have to do six to get down to one I have to divide that by six So I have to do the same to the other side Divide that by six which in to give me at four over three Now I need to get from one to twenty four to get from one spoonful and these times that by 24 So this is same as the other side Times by 24 so four over three Times 24 equals 32 I know it’s different ways you can do this and you [can] see this one has included a Fraction you can also be asked to work out Parts these need to work out the total number of parts in this case which would just be 8 plus 6 Percentages can come up on the calculator or the [hmoNg] calculator paper So it’s worth knowing a couple [of] ways to work them out online how to enter paper you can work them out by using a blocks [or] tens fives and ones so 10 percent of 824 will be eighty two point four Five percent will be half of that which would be forty one at point two and then one percent of that will be eight point two four and Using [correlations] of ten fives and ones you can generally make any number that you like so in your calculator paper You need is trying to 32 percent of it or you can do is take 32 divided by 100 come up with a number in this type into calculator which is 0.32 multiply that by 824 and You can get your answer like that Interest is just Another way or another word for working out percentages So if you leave your money in your bank the bank will [pay] you interest on it unfortunately [not] very much interested moment It’s going to be these three [percent] interest or 800 people counts how much would you get after year? If we have questions or equations using direct or inverse proportionality for direct proportionality We can say that x is equal to y, but we need to put a constant in there Which is Generally K and? You can take any values that are given to you the question plug that in find the value of K and then further use value Okay, and any other questions for things lit inversely proportionate. We can say that x is equal to k over Y again pop any values in work out the value of K and then you can go on to use that value of K to finish of the question When things are directly proportional you are going to get a nice straight line graph When things are inversely proportional you are going to get a curved graph When you have a graph the gradient or graph is going to tell you to show you How quickly how things are changing or the way to change the example? I use here is distance time graph and speed the rate of change is going to be distance divided by time So our gradient is going [to] be the changing up divided by the change in a cross There are two ways you can work out interest the long slow and methodical way or you can remember the equation So the long slow methodical way is to work out each year by year by year so if we start with 795 pounds at 3.5% compound interest and at the end of year one that they are going to have a 795 pounds plus the interest which is acute which is going to be twenty seven point eight two five which is three point five percent of 795 smokes going to equal eight hundred and twenty two eight [two] five in total that is the number you then use For the start of next year work out three point five percent of eight hundred twenty two which is going to be twenty eight seven at nine and this real number star getting quite a long So you need to round them or you [install] them in your memories calculator? But don’t around them too short Otherwise you’re going to get a rounding error and what you’re going to end up in you having the wrong answer for either white your entire number down from the calculator or Store it in that your limb ill calculator memory So [after] three years we’re going to have eight hundred [and] eighty-one pounds and forty three pens Now the equation that you can remember for this is The final amount is you could pay open brackets one plus R/100 close brackets to power of N? Where p is the amount that you start with R is interest Rayon N? Is the number of times you’ve acute interest which generally is going to be a year by year? So popping that in here. We will see that. [I’ll] start around with 795 One Plus three point five Over hundred [^] [3] gives us really really nicely the same answer One [ways] Quicker, but you have some [remember] the formula the other ways slower and is more likely to make mistakes [because] you have to write [down] a phone [number] and if you do it the way that these demonstrate want you to do it and Someone if they want use formula you get a year-by-year you might lose your marking [your] working marks so this is a tricky one you need to be careful of If you’re doing and foundation take you have now finished this unit you can even on to the other units and check what you do Don’t know in the revision guide, or you can go and track the exam papers Hiatus students, don’t worry. We don’t have too much left to do You need to have to use a tangent to find the gradient at point on a line because in real life it is very Rare that we actually get straight line [graphs] so here We have our curve line and we want to find the gradient the rate of change at this point here So what you need to do is you need to get your ruler And you need to draw a straight line exactly at that point you will see that this Line here is not the gradient for any other point on this line and then from this straight line that you have done you can then use that to work out the gradient of The line when you’re working [out] the gradient of a line bigger triangles the better, and we need to do changing up by changing across For iteration we are looking for the closest approximate solution so generally if I give you an equation that you can’t solve by any other method and tell you that the value of employing this case x is between 7 & 8 So [those] [images] don’t have a nice neat table see making everything to examine exactly what you’re doing Start by popping x is 7 there start walking x is a [in] [there], so you’re working [to] work out What it actually is and then make a comment, whether it is? too big Too small and then try intermediate numbers if you’re really really close only [go] up a little bit if you’re very very far away Go up a big bit if you’re not sure we’ll try a number in the middle and there is a little bit of luck involved in this because depending on where you start and Which path you go you may get to the solution quickly, or it may take you a little time to get to a solution? Some nice definition starts off here point is going to be the point also Shape where two lines meet a line is going to be [thin] connecting two points [though] sees that is going to be the point or on a lip diversity is going to be the point of Steepest Angle an Edge again is along here And then the plane is going to be a flat surface that goes in all directions Here we have some parallel lines here. We have some Perpendicular lines and a right angle can be indicated if we just draw a little box in the corner there Parallel Lines can be shown by little arrows going like this and if we look at Our equation for a straight line Parallel lines going to have the same gradient so m is going to be the same on both of them For a regular polygon. We’re going to have all citing same and all angles being the same and if n is our number of sides We can find the sum of the interior Angles, but in n minus 2 times 180 This applies whether it is a square or regular pentagon like your hexagon with a hexagon Octagon, [Dongola], ETc Etc When we reflect something we can flip it or reflect it across [a] line creating a mirror image Rotational symmetry is when something looks the same as it did at the beginning after you turn it round little bit if you want to draw a perpendicular bisector Of a line you need to even if I’m pushing [it] to open a compass really wide and we’re going to start by putting a point of our compass on a Over here, and what you need to do is draw a big circle so a big Arc That goes around like this Switch your compass points is over on B again draw a big Arc From B. And then this point and this point here all the points that are important. You need to join these two up with a pencil and a ruler a sharp pencil a ruler on [a] compass obviously I know I’m not doing that but you need to do that in your exam and Then this line here Is going to be your perpendicular bisector? for lesser problems you may be given a Scale that you have to use So you’re going to need to set your compass and your ruler to [that] scale or you may not be given one what you’re generally going to be asked to do is to draw a circle [around] point a with a compass not roughly like I’m doing and then a circle around point B again with a compass not like I’m doing and See where those two intersects, but the really really important thing here is Accuracy with compasses accuracy with rulers so if your window at the moment is covering tip x is broken The markings will come [off] it Please please please go and spend 58 pounds getting yourself a decent clear Ruler or if you can’t do that Don’t talk to your head a year or your form tutor or your math, or science teacher And they will generally be able to help you If we have angles that all meet at point a plus b. Plus C [plus] d are going to equal? 360 if they are on a straight line a plus b is going to equal 180 if We have vertically opposite Angles They are equal When we have Parallel lines are? Corresponding Angles are equal don’t forget this is also going to have alternate Angles and they are also going to equal and It is going to have alternate Angles now the example does not like you calling these Red Angles, but they are also going to be equal So if we could use the terms corresponding Angles and Parallel lines alternate Angles and Parallel lines vertically opposite angles and just don’t call them by a local terms like their angles and All the Angles inside a triangle add up to 180 degrees A Square is a regular quadrilateral is going to have equal sides it’s going to equal angles these are 90 degrees these are right angles and It’s area [you] [see] two thousand multiplied so that’s going to be x squared for this primitive for x and then for dragon ones can be x times square root k a Rectangle is going to have two sets of equal size. This is going to equal and these are going to be equal But the Angles inside are going to equal again. They’re going to be right angles It’s area is one side times the associated length times width parameter is 2x plus 2y a Parallelogram is legislative skew [with] triangle or rectangle for the parameter means group will go around the edge stays 2x plus 2y for the area we need the base times the height is going to have a two sets of Lines that are the same the opposite Angles are going to be equal and angles a and endless be going to add up [to] 180 For a trapezium, we’re going to have four Random sides with for the [perimeter]. [it’s] even speed. Let’s see this disco ball there on the edge that area is half The top of the bottom times the height and then the top and the bottom [I’m] just [going] to be parallel for a kite We’re going to have two sets of lines that are the same so our perimeter is going to be 2x plus 2y our area is going to be half a Plus B. Sorts of height times the width and we’re going to have one set of [angles] that [are] the same for a rhombus you’re gonna have all sides being same to the perimeter equals 4x the area is going to be half a Times B and opposite Angles [are] going to be the same If you want to prove that two triangles are the same you need three pieces of evidence And there are [4] different ways you can get three [piece] of evidence side side side side angle side angle side or right angle hypotenuse side So these two triangles is another drawn in different orientations. Have three sides that are the same so they are the same? For side angle side the Angle has to be the angle in between [two] sides So here we go side angle side side angle side. They are the same Here we have two triangles that have two identical angles and a side of same cities are the same our Last two triangles both have a right angle. They both have same hypotenuse and one of the other sides is the same For an equilateral triangle. We are going to have all sides and all angles being the same for [our] sausage triangles. We are going to have two sides being the same and Then the Angles at the base are going to be the same if you want to rotate to shape the examiners May even tell you what to do or they may have a rotate the shape and also you watch D You need to give the degree that it’s rotated by the direction It’s rotated by and where it’s rotated about so you’re going to rotate the shape 90 degrees Clockwise about 0 0 and The first thing you need to do is find the first point Which for me is this point here? Rotate it 90 degrees, and then all the other points will follow on from that If we’re going to reflect the [shapes] that will generally give you an align to reflect it in work out How far away your first point is from the line and that will give you your second point that? Will give you where your first point easier slow the line all your other points can reflect on a from that If you were given a translation today it may come as a vector so hearing is translate by [3/4] that means every single point has to move three crossing x and then four up in y So we are going to end up with our shape Up here somewhere If we are going to enlarge a shape you [need] to know the centre of enlargement your lines from the centre of enlargement through each of your Corners and then work out this from your center to your first corner Mm since I say enlargement is [two] and then that will be where the corner ends up and Then your shape draw and slightly better than mine will end up the larger and further away from your central larger ones here is the center [of] our circle the radius goes from the center to the edge a chord is a line that touches both Edges a diameter goes through the middle and touches both edges circumference is distance all the way around a tangent is a straight line that touches the circumference and ARc is part of [a] circumference [executive] is like a pie shaped wedge of a circle and the segment is cutting a bit off of it when we are talking about cubes or cuboids you need to know the places the Edges the surfaces and the vertices For a cube it is going to be identical on all edges which means all six spaces are going to be identical all Twelve Edges really identical is going to have eight vertices if we have an Edge which [is] of Length x The volume is going to be x cube is base times height times width and the surface area is going to be six because there Are going to be one two [three] and three behind six faces times x squared? Okay, way slightly different because each of the faces are not identical prisms and cylinders can be sort of of 3d versions of shapes that we are very familiar with in 2d if you want to work out Surface area these sometimes it helps to think of these as flattened out shapes, and then it becomes a lot less intimidating Because working out the surface area [of] a cylinder can seem intimidating [but] if you break it down into a circle and a large rectangle, well, that’s no problem at all and To work out volume we need to do surface area times the height For pyramids cones and spheres the best thing you can do in the exam is to slow down and think Logically about things how many phases can you see on the pyramid and then don’t forget about the ones behind? the frames of the cone and the sphere Turning a 3D shape into a 2d shape or 2d. Shape into a 3D [shape] can be incredibly Incredibly hard to do. This is a really really tricky skill. I probably didn’t [want] you to either draw three ovations So maybe top side in front or they’re going to give you three and ask you to draw the 3D shape this can be an Incredibly hard skill if you are not sure about it [if] you are having trouble visualizing things in 3d, [or] If you’re just not sure about how things are supposed to look the [best] thing I can suggest you to do is to use what you have on your pencil case and stuff that you’re allowed to have in your pencil case like your Sharpener and your rubber and try and use those [to] maybe try and construct some kind of 3D Shape that looks similar, but this is this is a mental skill that you have to do in your head. It’s Correlated if you are not used to it and the best thing you can do is practice with this one for a maps scale conversions Bearings is really important that your pencil case is up to date So I’m talking a clear ruler that you can use that has the divisions on it Not one that’s [covering] tip x was broken the same with a protractor and same with a compass So [what] they could ask you to do is give you point a and point B Ask you to measure the distance between them, and they want centimeters. It’s three point five kilometers turn that into a conversion They might also ask you what the bearing is So how far [around] it goes from north for that you’re going to need your protractor? You need to know and use all these formula now they start off by saying you needed to know all [of] them now They said that some of them. They will give you in the exam. So here are the ones that you need to know volume of a cuboid That is length times width times height [for] volume or prism [they’re] the area of the cross section times the length Pythagoras a squared plus B. Squared equals C squared trigonometry when it’s fine Those are opposites over hypotenuse the area of a triangle That is half base times height the area of the Trapezium That is half a plus B. Which is double comments’ times the height [of] the programs in the circle when you have a radius 2 times Pi times radius The area of a circle that is Pi times the radius squared Sub C. I’m giving it away and giving it like the area of a rectangle Length times width the area of a parallelogram is worth that base times height The volume of a cylinder that is PI times the radius squared times the height Circumference the circle when you have the diameter that [is] pi times diameter Trigonometry when it’s tan that is opposite over adjacent trigonometry when it’s Cos That is adjacent over hypotenuse Speed is equal to distance over time Density is equal to Mass over volume and Pressure is equal to force over area To help you with this if you want to save it time you can download all those flashcards from my website To work out the length of an ARc what you need to do is to work out the circumference and times that by the ratio What I mean by that is this conference call thing so 2 pi r the whole thing is 360 how many degrees round was [the] [circumference] so 2 Pi R times n over 360? To work out the sector area is very similar you need to find the area of the total circle and then you need to times that by Bill ratio so area of total circle is going to be pi R squared if This is N Degrees or realms there is going to be pi r squared times n. O 360 remember all the way around in a circle is 360 [halfway] around is 180 when we have similar Shapes the Corresponding sides are going to be in the same ratio as each other and the corresponding Angles are going to be equal [for] right angle triangles you need to be thinking pythagoras and you need to be thinking Trigonometry, so for right angle triangle. [we] have a and we have b we can write as C [by] using the formula or If we can work out one of the other sides or an angle now the really important thing here is Labeling your angles if we have angle the one opposite living aside So an opposite is opposite one next to it is the adjacent and this is the formulas here I’m sure there are loads and loads of Brilliant ways mnemonics that you have remembering those so please please [please] share them down in the comments to help other people I Strongly recommend you spend some time learning these values they can come up in lots of different ways to help you I’m going to show you the video or you can save it time and get flashcards for my website so sine of 0 is 0 Cos of 0 is 1 tan of 0 is 0 Sine of 30 is 1/2 Cos of 30 [is] Square [root] of 3 over 2 tan of 30 is 1 over Square root 3 sine 45 is 1 over Square Root of 2 Cos 45 is 1 every square root of 2 tan of 45 is 1 Sine 60 is Square root of 3 over 2 Cos of 60 is a half tan of 60 is Square Root 3 sine of 90 Is 1 cos of 90 is 0 and tan of 90 you don’t need to know? When we are talking about vexes it is excellent Op and y on the bottom so if you want to apply translation It is [a] movement in x and then the movement in y if we want to add subtract or multiply them it’s just [like] doing normal mass just slings on top of each other, so [if] we want to add vectors a and B First of all we look at the top line so 1 plus 3 is going to give us 4 then we look at the bottom 2 plus 4 is going to give us 6 Subtracting them is the same so 1 minus 3 is going to give [us] minus 2 and then 2 minus 4 is going to give [us] minus 2 Multiplying them works in the same way 2 times 1 [is] 2 2 [times] 2 is 4? If you’re doing foundation [tear] well done you have finished you can move on to the next section you can try some questions Or you can try some past papers to help you apply what you’ve learned When we are realizing my negative scale factors It is very similar to enlarging But positive ones you just need to go in the opposite direction so again you need to draw your lines from your Corners about your center of enlargements and Then here if this is one away [in] this direction We [were] lodging it [by] -2 [so] we need to go the other direction in minus 2 So that will go there, and then all the other ones will follow on like at A combination of rotations reflections and translations is Going to be tricky, but salah Swings through logically in of the exam take things carefully Look at what has [happened] to it and describe absolutely everything that’s happened. Don’t skip any steps circle theorems can see nasty by each need to read levels an Angle at the Center is going to twice the angle at the circumference Angles that [are] in the same segment are equal the angle in a sensor who is a right angle opposite Angles in a circle quadrilateral add up to 180 degrees [a] Call can be bisected by a perpendicular line [from] the center The angle between tangent and the radius is 90 degrees to tangents that come from the same point are equal and Alternate segments clearly tells us that this angle and this angle will be equal and this angle and this angle are going to be equal Here are the last three rules that you need to? Learn and be able to apply so we have the sine rule with the cosine rule and the rule for working out The area of my angle triangle, so for this [final] you need an angle or two sets of Angles and corresponding sides for cosine rule you need an angle and then three sides and you can find where the size or the angle and The error is an on right angle triangle. You need an angle and then two sides in an exam you won’t be given a set of raw data and from this you might be expected to draw a Graph you might be expected to do some maths work out the mean Median or the mode [you] might be expected to draw slightly more complicated gravity to draw box plots and this video is going to cover all [of] those skills patience Tables are one way of interpreting information and If you’re given a raw piece of information you may be asked to put it into the table. So here’s always me to the top even number Odd number and you start off by setting this in [Italia], So one When [I’m] not two three five five seven nine Eleven Eleven Eleven Twelve Thirteen eighteen Bar charts are another way of doing things and what you need to be really careful with bar charts is unity is a pencil and a ruler you [need] to have your balls neatly you need to have Titles you need to have units. You need to have a key part out I touch more complicated to construct because we need for everything into 360 degrees so here we have everything out of a hundred We need to work out [what] each degree is going to be worth [ministry] 360 Divided by a hundred that gives us each is worth three point six Where they need to take 50 [percent] or three times that’s 50 times Three point six is going to give us 180 degrees for strawberry ten times three point six is going to give us 36 degrees the chocolates 14 times three point six is going to give us 144 degrees for Both of them you need to draw a circle, please [le’veon] a circle not by [hydral] then measure 180 degrees and Remember label things It is really important that you label your sections Pictograms can be used something similar so say we were using strawberry chocolate or [both] again and We had a key with one triangle equals 10% strawberry at 50% would get five triangles chocolate at 10% would only get one and then both at 40% would get full triangles for this one. They might ask you to fill in a missing Section or to work out the key or to work out the total usually to use logic and the information that you have been given with line charts again for your axis you need to have title Units, and you need to have a key and the inch point mark on With a neat little cross with a sharp pencil where you want it to be? If you are going to be comparing two sets of [eight] [or] graph. It helps have line through them, so We could have this ylim This could be time Fiscally velocity and we can pair the acceleration by little at the gradients of two of these Or it could be a distribution graph and we can compare the distribution based on where the Peak of the graph would be to calculate the median it is the middle number, so if we Look our numbers Work [it] won’t have to be in order for this work in From outside we can see the median is nine. The mode is the most common at number and For that you can see eleven turns up three times the mean is the values added together and then divided [by] the number of values, so if I add all the values up let 128 there are 15 values 15 divided by 128 is 8.53 [reoccurring] the range is the highest number which is 19 minus 1 which gives us 18 The [radial] class is worked out from a table, and it is the class with the [most] Common numbering so in this case it will be this one here for above 10 There are five types of gas graphs you need to be familiar with the first one over here is showing no correlation Then because all the points are tightly packed we have a strong positive correlation Here the points aren’t very tightly packed so we have a weak positive correlation here again that the points are tightly packed But they are going down, so [there’s] a strong negative correlation And then here we have a weak negative correlation now using a pencil and a ruler You can draw a line of best fit through these things please use pencil and ruler not like this, and you draw a line of best fit for these things and Fond of the lines that you’ve drawn on your grass you can Extrapolate these lines forward into the future and make predictions about what is going to happen just [so] well out there and I can edit out There’s a very particular way the histograms need to be drawn your groups need to be your x axis So everything along here say this was one this would need to go along here like this and They do not have to be the same size So here this second one is Larger than the first one and again the third one is larger than the first and second one the frequency is the [number] of Things that go in that group, and then the y axis is a frequency a team that [needs] to go up the side here Knows the first [thing] that you need to calculate the frequency density and that’s calculated by frequency divided by class width I Really like box lots and a good tip [for] the exam is to sketch one out quickly Write the numbers in once you work the numbers out and then [draw] it properly on to the grid that I give you So over here. We have the lowest value Over here. We have the highest value the box ends at the lower Quartile and the lower quartile is N plus 1 [and] n being the number of Values divided by [4] the Median is 2 N Plus 1 divided by 4 and the upper quartile is 3 N plus 1 Divided by 4 the interquartile range is here And that’s just that upper quartile minus the lower quartile I definitely recommend you sketch out size before you draw it out neatly on the exam paper I Can describe the probability of an [ailment] from a table or a [tray] so here on table? I have some results [and] I’ve got the number of times that each of these results occurred to find out the permeability of any particular Results occurring you need the number [of] times That be the number of times it occurred over a total so here the total is 100 if I want to find the probability this three occurring about [B] 19 over 100 a 5 would be 27 over 100 and so on and you can use tree in a similar of the same way Looking at the number of times inviting it by the total number of times it could have possibly happened. [I] Can determine if your event is fair random or equally [unlikely]? So an exam question [there] might say something like a fair diet is thrown this way you can assume that each event is equally likely Or if it’s a way to die, they should tell you how is weighted Alternatively they might [expect] you [to] [be] [able] to work it out from a table now If you’ve got a very low number of total times we can say that’s a [low-end] number it’s going [to] be very hard for you to tell but if you have the more results you have some more times the dice or Whatever’s been thrown we say there is a high N Number and it’s much more likely to even out so these numbers here if we threw this dice or spinner or counter and A Thousand times would be much more likely to eat them out if it was a fair Spin up if it was a weighted one then [that] the weighted would show up more often the more times. It was thrown I Can give value out of one for the probability of an event? So if you go back to our first [such] data the probability of getting a five is Twenty seven over 100 we can write that as a fraction or we can write it as a decimal so [not] print to Seven and that is a value out of a hundred to give the probability of an event now all probabilities Equal one because when you throw a dice you are going to get a number doesn’t matter if [they’re] [divers] and even [divers] a weighted dice or Whether it’s a six dice if you throw the dice you are going to get a number so all probabilities will add up to [one] you might be asked an exam to combine set the probability so say we had a dice and A coin and you’d have to work out all the different Combinations, so for this you’d need to draw a table so your dice your coins your die fat obviously have one two three four five six and Different ways of landing one two three four five six your coin would have two different ways of landing so heads or Tails and then we have a total over Twelve Possibilities, so that you can work out The individual probability of anything so the chance of getting a four and a head is going to the one out [of] Twelve Venn diagrams can look scary, but they are not that bad you just need to get used to the way of naming things so here in our little Curly brackets is a set of data and I’m going to call that set a and element Not the chemistry elements. I’m afraid is part of the set So we can say that [four] is part of set a the number of elements Is going to be the N number so we can have n of a and in this case N? Equals six because that’s the number that I’ve gone up to each part of the set is a member So we can say that six is a member of A and then all items in the set [are] Everything so we use this symbol here To say everything in the set now when we have our venn diagram we might be combining two sets of data so here a set of data A and Here is set of data B. And if we want something that is in both sets of Data so in a and b we can use this in the mixture [a] and B if we want something a B that [is] in either a or b We go a or b And then if we want something that isn’t in our set of data Here is a [he] [was] everything else he is Everything that Is called a everything else so the bit outside our diagram is a everything else The new month sure [here], [you] can make these a bit nasty, but it’s still just Proviously still treat it like any other [probability] and you [will] be fine Whenever you see a word question probability try drawing a trait. So here we have two dice being thrown so first ice second [eye] And I want to know prohibiting an odd number or an even number so odd even odd even odd even Now we know that there were going to be six total possibilities one two three four five and six and that Odd is going to be 1 3 or 5 so it is 3 over 6 or 1/2 and even T46 is [only] 3 over 6 or 1/2 link put these on half half half half half half Now with these would air the you see a probability question what I want you to do is decide Which path you’re going to follow and then highlight the party were going to follow so here? [I] [will] be getting too old and here. I’m going to get into evens Rolling a dice twice and the kit results are both going to emit dependent events so if we want to combine these we look along our highlighted pass here and see we’ve got half and Half so we have odds write it down here 1/2 and 1/2 and and can be replaced with times, so this is 1/2 times 1/2 and then we do the same for even so is it 1/2 and 1/2 Giving us each m has a probability of [1/4] now if we want you to find out what our talents are getting odd and odd or even and Even we would then need to add these numbers Again giving us a half so and is times odd if class Or is plus [9] days time, or is [plus] in a bag we have [4] Red Balls [1] 2 [3] 4 and we have [2] blue balls After ball is taken is not placed well chances you pick two balls of the same color so again We’re going to draw a tree to work this out. This is your first pick Here is your second pick And we’re going to go red blue Red blue Red Blue now the important air bit information here is that they are not replaced So when you take that into account when we are drawing our probability tree so the first and they probably put [on] that red is going to be for a Total of [6] balls baloo is going to be 2 out of a total of 6 balls After we have taken [a] ball out the total is reduced by 1 so the total is now 5 balls in there, so all of these are going to be over 5 and this is the bit that people sometimes need So if I take your red ball out first of all there are now only three red balls But there is still two blue balls if I take a blue ball were out first of all there’s only one blue ball left But there are still four red balls left Going back. We’re going to answer the question now What are the chances you pick two balls the same color so reasoning to pick red and red or we can click blue and blue? Because your probability of picking your second ball is affected by what you picked in the first round this is a sector of dependent events so [radioman] red is going to be 4 over 6 and Which is times 3 over 5 and then blue is going to be 2 over 6 and? 1 over 5 our probability of picking red, and then red is 205 our [progress] we picking blue and then blue is [115] if we add those two together we will get a total bill to picking red and red or blue and blue of seven over 15 the next section is variety students only so if you’re doing Foundation teach and now going to look at some of practice papers go back and check through your vision guide Make sure you understood everything going to find some individual videos that you didn’t understand or go and watch the next whole topic video Conditional probability is going to be combining probability and Algebra, Algebra links in all over the place So [maybe] we had two events event a and event B and we knew the probability of event a Which is one like that? equals say not point three But we don’t know what the probability of event b. Is we want to find out the probability [of] event a and event b [so] probability a and B what we do know is the conditional probability so a probability of isn’t be happening given that a has already happened, and We can write an equation to work this out. So you can say the probability of event b given that’s what that line means a have happened is equal to the probability of A and B divided by the probability of Event a happening, so nothing too complicated [here] And then sorry once you’ve done that you can just take the numbers that they gives you in the question pop them in there with small numbers here pop them in and then you can work out the answer so it’s just a combination of Algebra and poverty nothing too complicated, but it might trip you up in the exam Whatever [making] it [zn] guys. This is quite a long video, but we’re going to take a second and be really dick This is only a vision you have done on the animation you’re planning on doing you’re not Going to be getting those top top grades [it] is completely unrealistic to expect two Three years with teaching to be squashed down into roughly an hour You need to go and practice practice practice practice [if] you want to be getting those top grades I strongly strongly suggest you don’t find a playlist for your example and Look at all the example work through the great nine playlist the great seven playlist Go and look at all of the questions and do as much practice as you can