Transcript for:
Math Exam Paper One Revision Guide

Okay, welcome to this video where we're going to be having a look at some essential topics to revise for the maths exam paper one. Now, within this video, we are going to be focusing primarily on a lot of the crossover content, meaning this video will be good for anyone sitting the foundation or the higher tier exam. It is primarily geared towards the foundation exam. However, a lot of the content is that crossover content that appears on both. If you are sitting the higher tier exam, there is another video which will be out and linked in this video to show you just the higher tier content. As always, this video will follow the usual format. So, if I show you a previous video so you can see what this looks like. When you are on the video, you can go into the description and there will be links there to download the booklet that follows along with this paper and the links for the foundation and the higher booklets that follow this particular set as well. If you keep scrolling down, you have also got all of the topic videos that accompany each of the questions in this video. So if you are stuck on any questions or you feel like you need a little bit more work on that particular topic, you can find the video lesson to any of them at any time that you want. As well as that, you can use the chapters feature and if you click onto the chapters, you can then scroll through the video having a look at whatever particular question you want to get to. particularly helpful if you have only watched through half the video and you want to come back to it at another time. You can just take a note of the topic that you're on and then you can follow it up at a later date. Now, hopefully you've got your revision under control and you're making great progress towards paper one. But if you're struggling to know where to start and need to maximize your maths in minimum time, then you should sign up for my rapid revision program upgrade as you can use it to build your very own personalized revision plan. Which means all you need to do is complete my revision quiz and you'll know exactly which lessons or topics you need to work on to maximize your maths in minimum time. This is my paid program, but you can sign up for a free trial on my website now. And if you feel like you need top tier support direction from me, then sign up for my live weekly interactive lessons which include revision specific lessons focusing on the topics that I expect to appear in exam papers one, two, and three. [Music] Okay, so looking at our first question. So this one says to work out an estimate and when we are estimating we have to round our numbers to one significant figure. So for 790 the first significant figure is in the hundreds position. So that will round to the nearest 100 to 800. 289 is closest to 300 with that first significant figure also being in the hundreds position. And then for 49 our first significant figure is the four and that is in the 10's position. So that will round to 50. Now clearly we need to work this out without a calculator. So we need to do an estimation method which is just multiplying those first digits. So 8 * 3 would give us 24. And then we're going to add on those four zeros that we didn't use. So one, two, three, four. We need to divide this by 50. Now when we are dividing, we can apply a little bit of a nice method here because we can cancel off a zero on the top and the bottom. And then for that division there, we're going to need to use bus stop division. So just to the side, we'll do five into 24 0 0, which is 24,000. Five doesn't go into two. So we'll go five into 24 which goes four times. Remainder four. Five goes into 40 eight times. And then we have our two zeros at the end. So our estimation for this calculation is 4,800. Now it should be noted as well you can do this more accurately than I have done. You could potentially round them to two significant figures. However, it would be quite difficult and you're not going to gain additional marks for that. So, it's always best to round them to one significant figure because we can apply those little methods like multiplying those starting numbers and having those zeros on the bottom when we are doing division that we can cancel off. So, there we go. That is estimations. When we are looking at product of prime factors, remember the word product means to multiply. So we are looking at the prime factors or the prime numbers that go into 124 that multiply to make 124. So to do that we start to break the number up and typically we use a prime factor tree. So we're looking for two numbers that multiply to make 124. Now it's an even number. So an easy choice would be to go for two and then dividing that number by two which comes out as 62. Two is prime. So we tend to circle that number just to indicate that we are done on that part of the tree. 62 is even again and we get 2 and 31. Now for this particular one 31 is not an obvious number as to whether it divides by anything. So it means it could be prime. Now you want to test a few numbers just to make sure it doesn't divide by two. It definitely doesn't divide by three because three goes up to 30. Now beyond that you want to test some of the prime numbers. So five is a good one to test and then seven and then a few numbers like 11 are also good to test. Typically in these sorts of questions if it doesn't divide by those numbers it's a good indicator that it might be a prime number. So testing seven. Seven goes up to 28 and then goes to 35. So seven doesn't go in. And hopefully some of the others we can spot that it doesn't divide. So 31 is actually a prime number. To write this as a product of primes, we just take those circled numbers and put a multiplication symbol between them all. If you want, you can write this in index form using powers. However, based on how the question's written, 2 * 2 * 31 is absolutely fine and you can write those in any order. But as there are two twos, we can write that as 2 ^ 2 or 2 ^ of 2 * 31. Either way of writing the answer is absolutely fine unless the question says you have to write your answer in index form. So there we go. That's product of prime factors. Now looking at a bit of a money calculation problem, we have a boat costs £14,200 plus 20% VAT. James pays a £5,000 deposit and the rest in 10 equal monthly installments. Work out the cost of each of the payments. Now, to start with, it does give us a massive hint there here that we need to add 20% VAT. That's what plus 20% VAT means. So, to start with, we need to work out 20% of 14,200. Now, without a calculator, the best method to go for is finding 10%. So 10% just knocks a zero off the number 1,420. Now for 20% we're going to need two of those. So essentially just multiplying this by two that'll give us 20% and that comes out as 2,840. We are adding that on to the cost of the boat. So really clearly showing that we are going to add that on. So 14,200 + 2,840. Adding those together and showing our working 0 4 10 7 and then 1. So £17,040. So that's the cost of the boat. And it says that James is going to pay a £5,000 deposit. So we need to subtract £5,000. Now we may need to use column addition for that. I can see that that's just going to subtract five from the seven in the 17 there. So that'll be £12,40. So that is the cost remaining that we need to pay for this boat. However, it's going to be paid in 10 equal monthly installments. That's a nice number particularly for a non-cal question because dividing by 10 to split this up into 10 months is a nice division. That's just going to cancel off that zero there. So each equal monthly installment will be 1 2 04 or 1,24. Onto some fraction calculations. When adding or subtracting fractions, you need to make sure you have a common denominator. Now in some questions, you may have to change both denominators, but always look to see if one of the denominators can turn into the other. And here we can multiply this right fraction top and bottom by two to make an equivalent fraction where they both have a common denominator of 12. So I'm just going to write next to that fraction. We're going to times the top and bottom by 2. And then I will have 5 over 12 add 22. If we add those together, we get a final answer of 7 over 12. Don't worry, although this fraction doesn't simplify, don't worry about simplifying a fraction unless the question says, like this next one, give your answer in its simplest form. If it just says to work out, it's absolutely fine to leave it not in its simplest form. You're not going to get any additional marks unless it's asking you to simplify. Now, for part B, we have a multiplication question. Could have a multiplication or a division. This one, we have a multiplication, which is possibly our nicest because all you have to do is multiply the two top numbers. So 3 * 5 which is 15. And multiply the bottom numbers which is 10 * 8 which is 80. So we get this final answer really nicely. But this one does say it wants our answer in its simplest form. So we need to have a look and think what these both divide by. One ends in a five, one ends in a n. So they definitely both divide by five. So if we divide the top and bottom by five and let's see what we get. we get 15 / 5, which is three. And 80 / 5. You probably want to take some time with that. The number five is an interesting one to divide by. It goes in twice as many times as 10 does. So 80 divided by 10 is eight. So twice as many times as that would be 16. But of course, if you're not sure, just to the side, do some bus stop division. Five goes into eight once, remainder three, and it goes into 30 six times. It doesn't take too long just to see you working out to the side. But there we go. Our final answer is 3 over 16. Just remember if we are gi doing a division as well, you need to multiply by the reciprocal when dividing. So you got to flip that second fraction over. But when multiplying nice and easy, times the top times the bottom. So we have a calculation here with some mixed numbers. And this time we are looking at a divide. So when we have a mixed number involved, whether there's one or two, we're going to need to convert that into an improper or a topheavy fraction. So an improper fraction, we do the big times the bottom, 1 * 5, add that extra 1/5, that gives us 6 fths. So rewriting the calculation with an improper fraction divided by 3 over4. And this is the scenario now where actually we have to multiply by this reciprocal of 3 over4. So instead of doing 6 over 5 / 3 over 4, we do 6 over 5 ultiplied by 4 over3. We can now apply that method of just multiplying the top. 6 * 4 is 24 and multiplying the bottom. 5 * 3 is 15. This question here says to give your answer as a mixed number. So to start with, we'll convert this into a mixed number. 15 goes into 24 once and there is a remainder of 9. So 9 over 15 left over. It also gives you a hint here. It says to give it in its simplest form. So you do have to check this fraction at the end to see if it simplifies. Now this one 9 and 15 they do both divide by three. So we can simplify that fraction and that would give us 1. 9 / 3 is 3 and 15 / 3 is 5. So our final answer there as a mixed number in its simplest form is 1 and 35ths. Onto reverse percentages in a sandwich shop 56 tuna sandwiches were sold. This was 40% of the total number of sandwiches sold. Work out the total number of sandwiches sold. So this question here, it tells us that the number that's given to us in the question, the 56 tuna sandwiches was 40%. If you kind of have a look and sort of skip out some of the words there, you can see it says 56 tuna sandwiches and then tells us this was 40%. So in this question, the first thing to do is to write down that 56 is equal to 40%. Now that means all the methods that we would previously use to find 10% or 1% have completely changed because if 56 was the original 100% then you can do things like divide by 10 to get 10% or divide by 100 to get 1%. But here because we've been given a smaller percentage we can no longer just divide by 10. If we think about this logically we want to get back to 100%. Now, if there is a number without a calculator that you can quickly multiply by to get to 100, we could do that. For example, if we were given 20%, we could multiply it by five. However, we are not. So, we're going to need to go down to a smaller percentage. Now, all we need to do is get to a percentage that turns into 100%. Typically, I like to go for 10%. However, in this scenario, might be nicer just to go for 20. I'm going to go down to 10 because it applies to more questions. So, if I go down to 10%, to get there, I would have to divide 40% by four. And then we can just multiply by 10. I need to do the same to the other side. I need to divide by four. Again, just to the side, I'd want to do some division. Four goes into five once, remainder one, it goes into 16 four times. So, 14 sandwiches would be 10%. We want to know 100%, so we need to times by 10. and we get 140 sandwiches. So there we go. Our final answer for this one would be 140. Looking at some standard form conversions with standard form, we need to put the number between 1 and 10 and then essentially just say how many times we have to multiply by a power of 10 to actually turn it back into this number. So to put the decimal, we want to have it in between the four and the three. That would make it 4.38, which fits in with our rules. That would be between 1 and 10. And to do that, we would have to do 1 2 3 4 5. We would have to multiply this number by 10 five times to turn it back into this 438,000. So that's 10 to the power of 5. And there we go. That's written in standard form. when we are looking the other way. So this 0 point number 0 0.562 it's pretty much exactly the same process. We need to put the decimal so it's between 1 and 10. So just here would make it 5.62. However this time when we say how many times we have to multiply it by 10. Clearly if we start multiplying 5.62 by 10 that's going to make it a larger number. So the only thing that changes is the power is going to be negative. So I can put the negative in the power and then I just need to count 1 2 3 jumps there for that place value and that would be -3. So there we go with a 0 point number our power is just going to be negative. When we are asked to convert back the other way it will say to write a piece of standard form as an ordinary number. Don't be put off by that word ordinary number. It just means a normal whole number or potentially a 0 point number in the case of this one where we are looking at a small number. So here we have 1.63 and it's a power of -3 with that 10 which means the decimal is going to move to the left and it's going to be a 0 point number. So when we are doing this I'm going to write the one, the six and the three and I'm going to write it slightly to the right hand side as I'm going to be jumping my decimal to the left. I'm going to put my decimal in between those numbers and then start to hop that decimal. One, two, three jumps. That's where the decimal needs to go. So, I'm going to drop it down there. And then underneath these loops, I need to fill in some zeros. And then just tidy it up with a zero at the start. Always rewrite the number, particularly when you're drawing these loops. So, 0.01 63. And there is my answer as an ordinary number. For part B, we have a positive power. This time, a positive power of three. So, I'm going to write the numbers one, four, five, and two. I'm just going to drop the decimal just above here. So, I can draw these loops and then go 1 2 3. So, it's going to drop all the way to the end, which means I actually don't have to write that decimal. I could write it as 1452.0, zero. But because it's a whole number with no zero at the end, I can just write 1 4 5 2. If I did have to do additional jumps, like for example, if that power was five, I'd have to do another two jumps and I would have a couple of zeros at the end there. But there we go. There's our two answers writing as an ordinary number. Now standard form calculations on a non-calculated paper are always quite challenging but we have one question here where we are multiplying and actually we can apply a little bit of a trick. Now what you could do is you could convert these numbers into ordinary numbers and then multiply them in a similar fashion as what we did when we were estimating. However, we can apply that method without actually writing them as ordinary numbers. So what I can do is just look at what that first whole number is going to be in both of these numbers here. 1's a 4, one's a six. So I can just do 4 * 6 which is equal to 24. So I know my number is going to have a two and a four in there. So I can write 24 and I'm going to need to times that by 10. And here we can just look at the powers. We have a power of three and a power of -5. Now when you are multiplying with powers you can add the powers. So here where I'm going to look at these powers I need to think about what they will add together to make. So I'll have 3 + the power of -5. Just getting those from here. 3 add -5 is 3 takeway 5 which is just -2. So actually my power here will just be -2. Now that does get me an answer. However, it does say in these questions to give your answer in standard form. Now, currently my number is not in standard form because it's not between 1 and 10. So, I need the decimal to go between the two and the four. And typically the way I remember this is if the number gets smaller, the power has to get one bigger. Now, in this case here, just be careful because the power is -2. one bigger than that goes up to -1. So my final answer there in standard form is 2.4 * 10us1. Now on the next question we have a multiplication and we have a division. This is a very challenging question without a calculator. It's a very very challenging one. Now what we could do is write these numbers in standard form and then sort of apply the trick that we've used above. So 0.06 06 is 6 * 10 ^ of 1 2 jumps -2 03. So we're going to times by 3 * 10 ^ of 1 2 3 4. So -4 and on the bottom we have 1 * 10 to the power of 1 2. So that's -2. So if we apply some of the methods that we have used in that previous question, we can actually complete the top there because we have 6 * 3 which is going to be 18. And then we're going to add those powers together. So if we add the powers on the top, we have -2 add -4 - 2 + -4 is equal to -6. So that's going to give me a power of -6. On the bottom we are now doing a division. So we have 1 * 10 to the -2. Now 18 / 1 is still 18. And then we're going to times 10. And when we are dividing just be careful here because we when dividing we have to subtract the powers. So we have -6 and -2 -6 subtracting the powers this time take away -2 when you subtract a negative that's going to be a addition so -6 add 2 is4 so my power there is going to be4 obviously you can see this is a very very challenging question so a difficult one that's thrown in here with these standard form calculations S. Now, it does say to give your answer in standard form. So, we would need to apply that little trick again, which is 1.8. To make the number smaller, we make the power bigger. So, -4 goes up to -3. And there's our final answer for that one. 1.8 * 10 -3. A very, very challenging one. Might be one that you want to watch. Again, onto some converting fractions, decimals, and percentages. We're going to write 0.23 23 as a percentage. Now any decimal can be converted into a percentage if you just multiply it by 100. However, I'd hope that you could kind of recognize this percentage here. But of course, multiplying by 100 would jump the decimal one two jumps and it would just become 23%. So I'm going to write there times by 100. But hopefully you could recognize that 0.23 is 23%. You just need to be very careful if it's something like 0.03 because that percentage is just 3%. When you times by 100, you'll see you just get the three. So just be very careful if you're using that method where you're just recognizing them. That applies a little bit here to our decimal question. So we have 7 over 100 as a decimal. So 7 out of 100. Not forgetting you could write this as a percentage if you prefer. 7% means seven per 100. So it is 7% and you could write that as a decimal like you did above or recognizing how we did it above you could divide that by 100 and that would give you 0.07. So you can write it in either form. We've got the percentage there and we've got writing as a decimal here. If we are ordering fractions, decimals and percentages, we need to convert all of these into the same format. At the moment, we have fractions, percentages, and decimals. I like to write them all as percentages or decimals. I don't tend to convert the decimals and percentages into fractions. I much prefer to look at them. And hopefully, you know that a half is 50%. However, if you don't, you could make the denominator out of 10. So, you could times it to make it 5 over 10, timesing the top and bottom by five, and then hopefully you spot that as 50%. For the next one, we definitely want to convert this so it's over 10. So tsing the top and bottom by two would give you 4 over 10, which is 40%. 0.55 is 55%. And from here, we can actually order these. So looking for the smallest, we have our 40%. And writing the original piece back in the order. So two- fifths, we then have the 45%. We then have the 1/2 which was 50% and finishing off with the 0.55. So there we go. Just making sure that we use a common format to actually put them in order for writing the original numbers back in the answer. Okay, so looking at some negative numbers, it says here in Iceland last year, the lowest temperature was -15° and in Iceland last year, the highest temperature was 42° greater than the lowest temperature. Work out the highest temperature in Iceland last year. Now, if we think about this in terms of a number line, we have -15° being our lowest. We don't know what our greatest was just yet, but we know somewhere along this number line is going to be the zero. Now, in order to get to the zero, we've got to go up by 15. So, we want to know how much further we have to go to increase by 42. So, what we could do is we could take away 15 away from that 42 just to see how much higher we need to go. So, I might do 42, take away 15, just some column subtraction to the side. I can't take away five from two, so I'll have to borrow. Five from 12 is going to go down to seven. And then one from three is two. So we have to go up an additional 27 degrees. Now if we go up an additional 27, which would be an overall increase of 42, of course that's going to get me to 27°. So here the highest temperature in Iceland last year would be 27° C. And there we go. That'll be our final answer. looking at this negative number question. Moving on to some algebra, we're looking at index laws. So with index laws, and we've already mentioned some of them when looking at standard form, when we are multiplying, we can add the powers. And when we are dividing, we can subtract the powers. Now, in this first question, we are dividing. So we're going to be subtracting powers. However, you need to be a little bit careful. Now, we have some whole numbers at the start of each expression. We have a 32 and a four. Now with whole numbers, all we do is divide those numbers. So we don't apply the power rules or the index laws when looking at ordinary whole numbers. So here 32 / 4, we want to work out either to the side or you might just know that that is equal to 8. So 32 / 4 is 8. And now we can look at these powers. I'm going to write the Q and I'm going to write the R. We just need to figure out what the powers of Q and R actually are. So here we have a power of nine for the Q and we also have a power of three. So if we subtract those powers, 9 takeway 3 gives us a power of six. Now on the next one, we need to be a little careful because although we have this power of four on the top with the R, we don't have a power written for the R on the bottom. When there's not a power written, don't forget there is a secret power of one that isn't written there. So just remember to write that in whenever you see a letter in these index law questions where there's no power. So here now we can subtract those powers. 4 takeway 1 gives us a power of three. And that is our final answer. Now that secret power leads in relatively nicely to our next question because the n in this expression here doesn't have a power with it. So I'm going to put that power of one in straight away. Now when we have brackets involved, technically what's actually been written down here is the power of three for everything inside that bracket, which means we would have to do 5 n ^ 1 p ^ 3 multiplied by itself three times. Now there is a quick way to get around this, but that's the long way writing it out three times. Now, that's not too bad because we can do 5 * 5 * 5. We can add the powers for the n's, add the powers for the p's. It won't actually take us very long just to write that down. However, that wouldn't be a very nice method if outside of the bracket there there was a slightly higher power. However, not forgetting we are working without a calculator. So, it is unlikely that you'd get a really large power there simply due to the size of the number that that five could turn into if we had things like power of seven and power of eight on the outside. So if we apply this method, we'll do the one with the three versions of it here and then have a look at maybe if we could spot a quicker way to do it. But 5 * 5 is 25 * 5 again is 125. If we look at the powers for the n, we have 1 2 and 3. So that becomes n to the power of three. And then if we look at the powers for the P, we have a three, a three, and another three. And if we add all of those together, we get P to the power of nine. Now, you can sort of take a slightly quicker approach from the brackets. All of these here, the powers are going to get multiplied by three. However, you got to be really careful with this because it's a bit little bit strange because the five also has a power of one with it. So it would be 5 to the^ of 3 n to the power of 3 p to the power of 9. When you multiply all those powers by 3, you just need to make sure at this point you actually do 5 to the^ of three which is as we've just worked out 125 and then n the^ of 3 p to the^ of 9. Doesn't matter what approach you take, but I do like this method of writing it out three times or whatever that power is outside the bracket just to make sure that we can see all of those powers that we're supposed to add together. Under some substitution here, it says p is equal to 4 x's + 3 y's. It now tells us that x is 5 and y is -2 and work out the value of p. Now the only thing we need to be careful of here really is that -2 because 4x which of course means 4 * x or four of the x's just means 4 * the x value which is 5. 4 * 5 is 20. For the next one we have three y's and y is -2. So 3 * -2 is the same as 3 * 2 but our answer is negative. So it's just -6. And then it asks us to actually add these together. So 4x is 20. We're going to add to that 3 y. We've just worked out 3 y was -6. So it's 20 + -6. And not forgetting when we are adding a negative that is just a subtraction. So we have 20 takeway 6 which gives us a final answer 14. And therefore the value of p is 14. Remember that word value refers to a number. So there's no letters that's going to be left in our question. A value is a number. So we have a final number answer and that is 14. Looking at some inequalities says here on the number line show the inequality and then we have this joint inequality here with different symbols in. Now, whenever we're drawing an inequality on a graph or a number line, all we're going to do is put circles above the numbers that we have in our inequality. Now, just to double check that there's nothing in the middle here because sometimes in the middle you could have something like x + one, maybe the same as this question, obviously not this question, but you could have something like that. And in that circumstance, you've got to get rid of this plus one and take away one from both of the numbers. However, in this one, we don't have anything to remove. So, we do have the numbers that we want to put the circles above. We have -3 and we have four. Now, in the case of a joint inequality, all we do is join them together. However, you do need to have a look and see if there's any extra, and I it's a little bit of a cheap method, but I just think about it as any extra ink on the inequality means extra ink in the circle. So if there's an extra line on the inequality, we also need to color that one in. And that colored in circle just represents when something is is greater than or less than or equal to. So if it's greater than or equal to or less than or equal to, we have that additional coloring in that needs to happen in the circle. The other one doesn't have the extra line on. It's not the equal to as well. So we don't need to color that circle in. For part B, we have an inequality that we need to solve. So in this circumstance, it's a very similar or pretty much the exact same process as solving equations. We want to figure out what x is less than. So to do that, we want to first get rid of this 27 doing the inverse by adding 27. When we do that, we get 7 x is less than 8 + 27 is 35. Now, this is what I would call a relatively nice one because when we now divide by seven, we do get a whole number. x is less than five. That's our final answer. Remember, if it's not going to be a whole number, you could just write it as a fraction. So, it would be strange to do so when it does divide, but you could write this as x is less than 35 / 7, which we know is five. So, we'll leave it as five. But if it doesn't divide, you can just leave it as a fraction. Something like this. It might also ask you to put onto a number line. Now, we do have the five over there. So, you would have a circle above the five. I'll draw it just below. And then you just point the arrow in the same direction as the inequality symbol in the question. Can see there x is less than five. These two symbols here are pointing in the same direction, which gives you a massive hint as to the direction of the arrow. But there we go. didn't ask us to draw that on the number line, but if it did, that's how we go about doing so. Okay, so factorizing quadratics. Now, when factorizing quadratics, really the important part here is to spot that this goes into a double bracket rather than a single bracket. When factorizing quadratics, you can have things like 8x + 20 and be asked to factoriize that. Now, when you look at those pieces there, you can see that they both divide by four. And if they both divide, it just means you can put four on the outside and then just think what do you times by to get 8x which would be 2x and what do you times 4 by to get 20 which would be five and you keep the symbols the same. And that can also apply and this is where sometimes people get a little confused because you can have things like x^2 let's go with minus 5x. Now that looks relatively similar to the quadratics that we are looking at. However, in that scenario, they both have an x in and there's no number at the end. And it's this number at the end here, which really changes it now into this double bracket because in that scenario, they both divide by x. You could put x on the outside and then have x - 5 on the inside, which is what you times by to get x^2 - 5x. So, that's not factorizing quadratics in the same sense that we're going to look at here. Because when you look at these quadratics here and you look at the x squar, the 4x, and the three, there's nothing that they all divide by other than one. And that's our big hint that we're going to need to use a double bracket. Now, with a double bracket, it is just the opposite of expanding double brackets. So here, we'd have an x in both. We're looking for the numbers that multiply to make three. Three is a prime number, so that's quite a nice one. There is only one pair of numbers which means that one and three have to go into the brackets. We just need to figure out what the symbols are going to be. The next thing that you look at is the number in the middle which is + 4. We just think to ourselves, how do we make plus4 out of these two numbers? Well, you would have to have positive 1. Add three. 1 + 3 is four. And there we go. That's our quadratic factorized. These numbers can be written in either order in the brackets. It's only really important we get the symbols right and quite commonly it's always the plus symbols but we'll have an example where we're looking at one where there isn't. We have another version down here. So doing the same again numbers that multiply to make nine. You could have one and nine or in this case we could also have three and three. So in this one we actually have a decision to make. So we have x in both of our brackets but this time we are trying to make plus six in the middle there. Now, you can't make plus 6 using 1 and 9. So, it has to be the threes. So, we would have x + 3 and another x + 3. 3 + 3 makes six. And they multiply to make nine. And there we go. We have our brackets factorized. Now, in this one, you do have two of the same brackets. You could actually write that as one of the brackets x + 3 with a squared on the outside, which just means the same bracket being multiplied together. And of course in this scenario remembering you could be asked to expand something like that which just means that you would want to rewrite it first as x + 3 and x + 3 and then you would want to go about expanding it. Remember if you're not sure if your answer's correct you can always expand it. So x * x is x 2 x * 3 is 3 x. You then move on to the bottom. 3 * x is another 3x and 3 * 3 gives you 9. And that simplifying process at the end where we add the two threes together is exactly why we're looking at the two numbers that add to make six because of course they make x^2 + 6 x now in the middle + 9. So lots of things that you can do to check your answer. But when we're factorizing we're just looking how we get it back into those two double brackets. Now, if we're asked to solve a quadratic, the process is pretty much exactly the same, particularly to get the large majority of the marks. Now, this one is a trickier question because we do have negatives involved, but we're going to follow the same process just because it says solve here. We are still going to factoriize first. It's just our solve part is one last step at the end. So, we have numbers to multiply to make 18. We have 1 * 18, 2 * 9, or 3 * 6. And in this case, we're looking to make -7. There's no way of getting to 7 or -7 using 1 and 18. So that we can't use. However, 2 and 9 have a nice link there. We can make 7 or - 7 using 2 and 9. So we know it's going to be this pair here. But we need to get the symbols correct. Now if we used pos 9 takeway 2 well 9 takeway 2 is pos 7 and we want - 7 which just gives us an indication that it's the other way round. So instead of positive 9 and -2 we want pos2 takeway 9 and 2 takeway 9 is -7 and those two numbers multiply to make -8. So the solve part well what do we do? Well, when we are solving, all we do is we look at these numbers in the brackets and we just flip the symbol. So, instead of having positive2, one of our solutions is x will equal -2. And for the right one, our solution will not be -9, but x will equal pos 9. Now, that's a nice little bit to remember. Essentially, all you're actually finding is what number has to go into the bracket to make it equal 0. Well, look at the left bracket. -2 + 2 is 0 and on the right bracket 9 takeway 9 is also zero and that's what we are trying to find where these equal 0 or options where they could equal zero. So there we go there's our two solutions and that is solving quadratics. Now looking at drawing nonlinear graphs now typically nonlinear graphs are a lot nicer than a calculator paper. However, if they do appear on a non-cal paper, we need to think about methods. And this is the same the same method that we would use if we were doing a linear graph or a straight line graph. It's just that our equation for this line or the it's just a little bit more complicated. So here we are told that y is equal to x^2 - 3x + 1. We have some values in our table. Now it makes sense to start with the positive values. So if I start with the four, I just need to put four in every position where there's an x in that equation. So x^2 would be 4^ 2 - 3 * 4. You can write 3 * 4. I like to write it like this with a bracket just to keep it a little bit tidier. And then + 1. And I just need to work that out and just be very careful as I move along. So 4^ 2 is 16. 3 * 4 is 12 and then + 1. 16 take away 12 is 4 + 1 is 5. I just need to keep doing the same again for all these numbers. It will take me a little bit of time but that's okay. We get a lot of marks for this sort of question. So if I move on to the 3, I have 3^ 2 which is 9 take away 3 * 3 which is another 9 and then add 1. So 9 takeway 9 is 0. add 1 is 1. Move on to the two. 2 ^ 2 is 4 take away 3 * 2 which is 6. So take away 6 and then add one. 4 takeway 6 is -2 add 1 is -1. And at this point here you might be able to see a bit of a pattern. So in this particular pattern look we have -1 next to each other. That's sort of the middle of our pattern. Outside of that we have ones and outside of that again we have this five and here will also be five. Now you can't always rely on that pattern. Sometimes you might need to do the negative number into your formula there. If we did we just got to be really careful because we have -1 squared. I always put it in brackets because that's how you have to type into a calculator. But it just reminds me that when I square negative 1, it does become positive 1. And I'm going to take away three lots of - 1. Really careful on this one because 3 * -1 is -3. And you're taking away -3, which is actually adding 3. And then + one. So 1 + 3 is 4 + 1 is five. And we do get five there. However, it was a little bit easier just to spot the pattern. It asks us to draw this. So, we need to plot these coordinates, being very careful that I put them in the right place. Remembering a quadratic curve will make a nice smooth U shape somewhere like this on the graph. If it doesn't, we've plotted the point incorrectly. We need to spot where that point is and fix it. So, if we plot them, -1 goes up to five, which is here. 0 goes up to one. One goes to negative 1. Two is on one. Three is on one. Four is on five. And you can see before joining this up, it is going to make that nice smooth U curve. Remember when you join this up, you are not using a ruler. So you're going to draw a nice smooth curve, making it clear that you are using a handdrawn curve to do so. And there we go. Doesn't need to be perfect, but it does need to go through all of those coordinates. If one of the points is out of place and it doesn't make a nice U curve, it does mean you've got one of those points incorrect. So do go back, check, see if you can fix that particular point so it makes that nice smooth curve. Now for the final part here, it says using your graph, find estimates for the solutions of the equation. Now these solutions, sometimes also called the roots, are the points where it crosses the x-axis. So here I want to very carefully read this coordinate and I want to very carefully read this coordinate. Now the one on the right looks a little bit easier to read. To me it looks like 2.6. You will be allowed a little bit of leeway either side of the number. So don't stress too much about getting it absolutely perfect. You don't need to write things like 25 2.575 or something like that. Just as close as you can. For the one on the left there, it looks like it's one square before the 0.5. You might argue just slightly in between. I'm happy with it just being one square before the 0.5. So 0.4. So the two answers that I would give, I would say 0.4 and 2.6. And there we go. There's my two solutions or roots. When looking at a quadratic graph, looking at a sequence, it says here are the first four terms of an arithmetic sequence. Write down an expression in terms of n for the nth term of the sequence. So when looking at a sequence, we are looking at what times table it's related to. So here having a look at what the jump between each number is. Now in this particular one, it's going up in sixes. So each time is plus six. Now the start of my nth term where we use the letter n is related to that particular pattern. So all we have to say is 6n that means the six times table. However, this is not the six times table. It's just related to the six times table. Now there are two different ways of looking at how to manipulate this expression to write what this sequence actually is. Now the method that I like is I like to think about the first number in the six times table which is six. And if I write it just above, it becomes very clear and obvious that this sequence is smaller than the six times table. Not forgetting the next number would be 12 and 18 and so on. But you can see the sequence that we've been given is actually one smaller than the six times table. So our nthterm expression is the six times table but take away one. And all that means is exactly as I've said it, the six times table take away one. Now that's the logical way of finding that number at the end. It's not always the nicest if you have a negative sequence which is going down. But in this particular question, it's quite logical. However, if you don't spot that, the other little trick you can use is to go back by six and you get -1. It gives you the number every time, which is a little bit nicer on some of the negative sequences. But there we go. That's our final answer. 6n minus one. Okay. So, looking at some unit conversions. So you do need to know things like how many millimeters or centimeters are in centimeters and meters just to make sure that we can convert these. There are 10 mm per cm and there are 100 cm per meter. Likewise you need to know that there are,000 m in a kilometer. So something like this 40 cm into millime well every cm has 10 mm in. So our unit here would be 1 cm is equal to 10 millimeters. So if we have 40 cm, we would want to multiply that by 10. And that would be 400 mm. When we are changing grams into kilog, the unit conversion we need to know is that there are 1,00 g in 1 kilogram. So here if we have 1,756 g we want to turn that back into kilogram. I'm forgetting that is dividing by 1,000. You can take a logical approach and think well it is 1,000 and a bit you know it's 1 something but if we just divide by a th00and 1 2 3 we get 1.756 kg. And again, if we were converting the kilograms back into grams, then we would multiply back to grams. So, our conversion works both ways. But there we go. There are just a couple of unit conversions. When looking at percentages of an amount, the nicest percentage to always find is 10%. From there, we can tend to get any percentage that we want. To get 10%, we just need to divide by 10. So we would get 16 g as 10%. This question here wants 15% of 160 g. So we need to go down to a slightly smaller percentage as well, getting our 5% which is just dividing our 10% by two. So 16 divided by two gives us 8. We now have our 10% and our 5% which combined makes 15% and that would be 16 + 8 which would be 24. So just remember when finding a percentage of an amount, find that 10% and then break it down into smaller percentages to get to the percentage that we are looking for in the question. When we are sharing in a ratio, read the question carefully because there are different variations of sharing in a ratio. But this one says Harry Regan and Keeland share £450 in the ratio 2 to 5:3. He wants to know how much Keelin gets. Now the important part to look at is how much the amount of money or whatever we are sharing is actually referring to here. It's referring to the amount that all three of them share. Not one of them, not the difference between them, but how much all three of them share. So it's related to all three of these numbers. Now if we add them together, 2 + 5 is 7 + 3 is 10. There are 10 parts in total. So to share this in a ratio, the first thing we want to do is to share it into 10 parts. £450 / 10 gives us £45. Now that is the value of one part. Now if we label Harry, Rean, McKeen and their numbers which are two, five and three. Now all of these need to be multiplied by 45. This question itself does only want to know how much money Keelin gets. So, we do actually need to only work out this one on the right. As long as we times that by 45, we will get the answer. But it is good practice to multiply them all. Now, 2 * 45 would give us 90. 5 * 45, take your time to work this out, but we get 225. And our final one, which is the one we were looking for, is 3 * 45. So, of course, make sure we get this one correct. We'll do our multiplication to the side. 3 * 5 is 15. 3 * 4 is 12 + the 1 is 13. So 135 is how much Keeling gets. That is the answer to our question. So it's 135. In some questions we may have ratios, fractions and percentages. And here is a good question here and they do tend to be a little bit wordy. It says a company has a total of 160 cars and vans in the ratio 3 to 7. says 1/8 of the cars use electricity, 25% use diesel, and the rest use petrol. Look at the number of cars that use petrol. Now, to start with, if we just focus on this first line, it says that there are 160 cars in vans in the ratio 3-7. So, to work out how many cars or vans there are, we're going to need to first split it in this ratio. It's quite a nice one here. They do add up to 10 parts, similar to a previous question. So when we split this into parts, we want to divide by 10. So 160 / 10 is equal to 16. If we label this up as cars and vans, to work out how many cars there are, we're just going to times both of these by 16. So 3 * 16, taking our time to work this out as well, 3 * 6 is 18. 3 * 1 is 3 + 1 is 4. There are 48 cars. We don't necessarily need to know the vans for this question because it now starts asking us about the cars. But let's just keep that really clearly labeled that there are 48 cars. Now it says 1/8 of those cars use electricity. So 1/8 of 48 is 48 / 8 which is equal to six. So six cars use electricity. Says 25% of the cars use diesel. 25% is a unique percentages uh unique percentage because it is one quarter. So you could write that as one quarter if you prefer or you could do your 10% 5% method to get your 25%. So here I'm going to work out one quarter which is 48 / 4 and that would give me 12 cars. So 12 cars use diesel. It says the rest of the cars use petrol. Work out the number of cars that use petrol. Well, in total here, if we add these two together, we have 18 cars. So, there are 18 cars that use electricity or diesel. There are 48 cars in total. So, for the amount that use petrol, we have 48 take away 18, which is equal to 30. So, the final answer for our question would be 30 cars that use petrol. And paying close attention to this final line here that says you must show your working. So it's really important along the way to show those steps like how we found 25% and so on just to make sure that we are showing our working and making sure that we are getting the full marks for questions like this. Okay, so in this question we are looking at percentage profit says Alice pays £10 for 24 chocolate bars. She sells all 24 chocolate bars for 50p each. Work out Alice's percentage profit. Now, particularly when you're doing this without a calculator, tends to be the percentages are quite nice numbers. So, let's work out how much profit she's actually made. We know she paid £10, but what does she actually sell them all for? Well, she sells 24 chocolate bars for 50p. So, in terms of looking at this as a an amount of profit, we need to first work out 24 * 50p. Now, that's in pence, the others in pounds. So, it's completely up to you whether you turn that into pounds first, but I need to do 24 * 50. Now, you might know what that is already. That's absolutely fine if you want to just apply a mental arithmetic method, but 0 * 4 is 0. And 0 * 2. Placeholder in for the five. So, 5 * 4 is 20. 5 * 2 is 10 + the 2 is 12. There we go. and we get £1,200, which if we convert that into pounds is £12. So there we go. She sells all of these for £12. I would then want to label or write down exactly what the profit is. So the profit here is how much she makes above what she paid. So she paid 12. Sorry, she paid 10. She sold them for 12. So 12 takeway 10 is equal to2 profit. Now we want to know what that profit is as a percentage. So in terms of finding that we want to do the profit which is 2 out of the original. So the profit out of the original. Now, if we write that as a fraction, that's 2 over10 or two out of the£10 that she spent. Now, looking at that, you might already know what that is as a percentage. If you don't, then we want the denominator to be 100. And to get there, we multiply by 10, which gives us 20 out of 100 or 20%. So our final answer here is 20% and that is is this person's percentage profit. Okay. So a ratio problem involving a line here where we've got a little bit of language to figure out along the way. It says the length AB is five times the length of BC. It then tells us that A to C, which is the full length, is 90 cm. Work out the length of A to B. Now this bit of language here where it says AB is 5 * the length of BC that is actually giving us a ratio. It's telling us that the left side from A to B there is five and the right is 1. 5 * the length is 5:1. So that little bit of language straight away gives us a ratio 5 1. Now the full length is 90. If it only gave us the B to C, that would be the one part. If it gave us the A to B, that would be the five parts. But this has given us the overall distance. So in total, we have six parts in this ratio. So we want to share 90 cm in six parts. Want to be careful when we actually work out this division. So we might want to do a little bit of bus stop division to the side for this one. So 90 divided by six. N 6 goes into 9 once, remainder three, it goes into 30 five times. So each part is equal to 15. We can then multiply both parts of this ratio by 15. The one nice and easy becomes 15. The five is going to be 75. So that's our ratio there. We know this side here is 75 cm. We know the B to C is 15 cm. Just need to answer the question. It does say work out the length of A to B. So the length of A to B would be 75 cm. Of course, she's paying particular attention to the language in the question because it might have told us the length of A to B, which in which case we would have divided by five rather than six because it would just be telling us the one piece of the ratio there. It might have also just told us the length of B to C, which case you just have to multiply by five to get it five times bigger. So, lots of different ways that you could be given a question like this, but the important part there is writing down that ratio for that language where one is five times the length of the other. And that particular piece of language can be applied in lots of different topics as well. You could have a quadrilateral where it's telling you that one angle is three times the size of the other. And you could write that as a ratio 3:1 and find any missing angles. So, lots of different places that that language can be included. There you go. That's how we'd approach this one. And onto combining ratios. Says here in a village number of houses to flats are in the ratio 7 to4 and flats to bungalows are in the ratio 8 to5. Says there are 50 bungalows. How many houses are there? Now in this scenario here you can see in the language that flats is in both of these ratios. But in one ratio it's represented by a four and in the other ratio it's represented by an eight. Now, in order to put these ratios together, there are multiple ways to solve these types of questions, but typically it's going to ask to combine ratios in other situations as well. And to make sure that we know a method for any scenario, you need that number that's in both to be the same in both. So, we want to find the lowest common multiple of four and eight. This is a relatively nice one in the sense that four can turn into eight. So, we want to multiply this ratio by two. If we multiply both of those by two, the seven becomes 14 and the four becomes 8. And now that eight is the same in both ratios. Now that it's the same, you can actually write a three-part ratio. So houses to flats to bungalows would be the ratio. Well, now 14 is for houses, flats is eight in both, and bungalows is five. From here, we can now figure something else out because it tells us there are 50 bungalows. That matches up quite nice with the five to get to 50. That is multiplying by 10. Which means I'll have to multiply everything by 10. And I'll know how many houses there are, 140. And I'll know how many flats there are, 80. So, where it says how many houses are in the village, well, we've just worked that out. There are 140 houses. And there we go. That's our final answer. Just having a look at that number, that thing that is in both ratios, and making sure that that number is the same in both. So you can put it all together and form what we call a three-part ratio. Looking at some direct proportion here in context, says that there are 30 children on a school trip. At least one adult is needed for every eight children on the trip. work out the least number of adults needed on the trip. So if we count up in eights, so eight children would be equal to one adult. Let's just write that as 1 a. The next eight would get you to 16 children and that would be two adults. The next one would be 24, which would get you to three adults. And the next one actually goes over 30. It goes to 32. And that would need four adults. Now, it says at least one adult is needed for every eight children. So, we can't have the three adults because that's only 24 children that are allowed. The fact that the next one goes over is fine. It just means that the final adult there is only for the six children. So, the least amount of adults that would be needed. We would need four adults. Now, it says in the next part here that two more children join the trip. Does this mean that more adults are needed? And you must give a reason for your answer. Well, actually no. You can see here that if two more adults were needed, that would be sorry, two more children were added, that would be 32 children, which is equal to four adults. So we' write some sort of explanation down, but the important part is we've shown the working. So we would write no because and then give our reasoning here which is 32 children is still equal to four adults. Okay. Looking at some speed, distance and time then. Always a very tricky topic to appear if it does appear on a non-cal exam. Let's have a look at a method we could use just in case it does. So, it says Carl leaves his house at 6:30 a.m. to drive to work. He drives a distance of 50 mi and Carl drives at an average speed of 40 mph. What time will Carl arrive at work? Now, there are different methods that you can use. You could write down your formula. Speed is equal to distance over time. You may or may not also use things like formula triangles where you have speed, distance, and time. But ultimately, if we're looking at this question, we want to look at that speed and think about, and I like to use a table method on a non-cal approach, just looking at it in terms of proportion. Speed is dealing with distance and time. And a speed of 40 mph means 40 miles in 60 minutes. You can leave it in hours if you want, but ultimately we want to get this to become 50 miles to see how long it will take him. Now, to do that, typically we can usually go down to 10. If you can't, you could always go down to one, but typically we can go down to 10. And if we get down to 10 miles and how long that takes, we can then go straight to 50. To get there, we would just divide by four. And we do the same on the other side. 60 / 4 would be 15 minutes. So therefore, that extra 10 miles above that speed of 40 mph would take an additional 15 minutes. You can probably see therefore that means it's an hour and 15. But to get there, we would multiply by 5. And 15 * 5 is 75 minutes. Again, just doing the same on the other side. 75 minutes is equal to 1 hour and 15 minutes. So, if Carl leaves his home at 6:30 a.m. and it takes an hour and 15 minutes, well, 6:30 plus 1 hour 15 is going to be equal to 7:45 a.m. And there we go. That's what time he's going to get to work. Says here, in fact, Carl's average speed was greater than 40 mph. How does this affect your answer to part A? Well, if he was going faster than 40 mph, it means he's going to get there faster. If he goes at higher speed, then my answer, the time will be reduced. So, I would write something along the lines of my answer being changing because it does say, how does this affect your answer? So, I don't just want to say Carl will go faster so he'll get there quicker. I want to specifically refer to my answer. So the time would be less something along these lines. So he would arrive earlier. There we go. That's specifically referring to the actual time there being earlier than 7:45. We may also have to deal with speed and time on a distance time graph. This one here says David drove from his home to the gym. He stayed at the gym and then drove home. Part A here is relatively nice. It says how many minutes did David stay at the gym? We just need to make sure we read it really accurately. So where he stays at the gym is this flat line here where he hasn't traveled any distance. So if we go down really carefully using a ruler, that is 12:45 that he got to the gym and then when he left it was 1:30. We got that 130 just here. So from 12:45 to 1:30, that's a total of 45 minutes. Part B here is a little bit more complicated. It says, what was David's average speed on his journey home? Well, on his journey home, which is this final part of the line here, well, we need to know how far he's gone and how long it took him to do so. So, if we go along here, you can see that that was 25 km. So, he's done 25 km and it took him 30 minutes. Now you might logically know what that answer is but if we want to write it down visually so we can actually see that we can write a distance and a time he's gone 25 km in 30 minutes. Now a speed would be given in kilome hour. So we want this to be 60 minutes which visually you can see now is just tsing by two. So the distance would also times by two and that means he's going 50 km per hour. So my final answer 50 km per hour is the speed that David's going in order to do 25 km in 30 minutes. Okay. So looking at some density, mass, and volume says here a cube has side lengths of 3 cm and the cube has a mass of 81 g. Got the density of the cube. Now again with density you might know density is equal to mass divided by volume. You might also use that in a formula triangle. All of these methods are fine. Density mass and volume. Now here to work out the density we just need to do the mass divided by the volume. We have a mass which is given to us 81 g. I just need to work out what that volume is. But ultimately looking at my formula here, the density will be equal to 81 / something. We just need to work out what this something is. So to get the volume of a cube, all we need to do is multiply all of the three dimensions together. And in the case of a cube, they are all the same length. So the volume for a cube will be equal to 3 * 3 * another 3. 3 * 3 is 9 * 3 which is equal to 27 cm cubed. So that's our volume. I just need to do 27 on the bottom here. So 81 divided by 27. I need to take my time to work that out. It's not the nicest. So 27 if we add another 27 that would get us to 54. and then actually adding another gets us to 81. So it wasn't actually too bad. We have three. Now in terms of writing the density, what we have just found out is that there are 3 g in every cm cube. There we go. There are there are 27 cm cubes that make up this cube and each one of them weighs 3 g and therefore the overall weight was 81 g. And there we go. there is some density, mass, volume. Now with column vectors, we can be asked to add, subtract some variations of them. We can al also be asked to draw them on a grid. Now this one here has given us two column vectors and it's asked us to draw and label the vector A minus 2 B. Now the vector A is given to us already 5 and 2, but we've not been given the vector 2 B. We've been given the vector 1 B. So to get the vector 2 B nice for us to find we just multiply the numbers in the vector by two. So 3 turns into 6 -1 turns into -2. It's asking us to take these vectors away from one another. So 2B now is 6 and minus2. And we just need to subtract these movements. A column vector remember is just a movement. The top number represents left and right and the bottom number represents up and down. So, we're just going to take these away from one another. 5 takeway 6 is -1 and 2 take away -2 being really careful because that's actually a taking away a negative which is adding two and that becomes four. So, min -1 and 4. Now, I need to draw and label this vector. So, I need to pick a point where I can go left by one and up by four. So, be careful where you put it. If I put it here, look, by the time I go left by one, I can't then go up by four. So, I need to sort of visually in my head think about where it's going to fit. So, I'm going to go down here. I need to go left by one, up by four. One, two, three, four. That's where I'm going to finish. So, left by one, up by four. Join those together. We put a little arrow on that line just to show the direction it's going in. I'm just going to label that A minus 2 B. And there we go. There's my column vector drawn on a grid. So this question here combines some plans and elevations and volume of a cylinder. Says here the centimeter grid shows the plan and front elevation of a cylinder. Now the plan is from above and the front is looking at it from the front. So hopefully we know how to draw a sketch of a cylinder. We draw sort of a squished circle. Two straight lines and you can just curve this line ever so slightly. So to work out the volume of a cylinder, the same as any prism, you have to work out the area of the cross-section and times it by the length that it goes through the shape. So in the case of something like a triangular prism, as an example, you want to work out the area of the triangular face called the cross-section and times it by the length that it goes through the shape. So, in the case of a cylinder, we want to work out the area of the circle and times it by the length that it goes through, which in this case, I guess we'd call the height when it's standing up. So, in order to work out the area of that cross-section, and it does say here, leave it in terms of pi, we need to know the formula for the area of a circle. So, the area of a circle formula, area is equal to pi r 2. So we need to know the radius of this circle which is half the distance from the circumference to the center. So here that is three. We know it's three because it says it's a cm grid. So each square there is 1 cm. We don't need to measure it with a ruler. So three if we plug that into p<unk> r 2 that would be<unk> * 3^ 2 3 2 is 9. So it's p<unk> * 9 and leaving it in terms of pi just means to write it as 9 pi. So the area of the circular face is 9 pi. We just don't have to type it into a calculator. So that's my area of the cross-section. To get the volume of the cylinder, I just need to times it by the height, which we can see here on the side of the front elevation. That's five squares. So it has a height of 5 cm. So to get the volume, I'm going to do the area of the cross-section, 9 pi, and multiply it by the height of five. 9<unk> * 5 is just 9 * 5, 45. And we'll keep that pi with it. So it's in terms of pi. And there we go. And there's my final answer, 45 pi. Typically, this would normally have on your dotted line something like cm cubed as it is a volume. If not, you'd also put your centime cubed with it as well. Okay. So, looking at some angles in quadrilaterals, it says here, pqrs is a quadrilateral. PST is a straight line. Find the value of y. Now, angles within a quadrilateral. We need to know add up to 360°. So, we need to add up the angle that we've got and see what's missing to find this missing angle here inside the quadrilateral. That's going to help us get to y as it's told us that PST is a straight line. So these two angles here which are next to each other on a straight line add up to 180°. So that's all the knowledge that you need to know for this question. If we add these numbers up then we use column addition 130 65 and 95. Adding those together 5 + 5 is 10. 9 add 6 is 15 + the 3 is 18 + the 1 is 19 and then we have 1 + the 1 2. So for the 360° we want to take away 290 which leaves us with 70° left over. So this missing angle is 70°. We also have just identified the angles on the straight line equaling 180. So to find the value of y, 180 take away the 70° next to it leaves us with 110°. So there we go. Y is equal to 110° and that's our final answer. Now angles in polygons are a bit trickier particularly when we are not using a calculator. There is a formula for angles in polygons. So typically a formula method is relatively quick to use. We take the amount of sides of the polygon. This does say it is a regular pentagon, but you can count the sides. 1 2 3 4 5 and we subtract two. So 5 takeway two is three. And that tells us how many triangles this shape is made of. Whenever you've learned this, you might have seen a little visual demo of splitting it up into multiple triangles. and it makes one, two, three, which we've just worked out anyway using our formula. We'll write the formula down in just a second. So now once we know how many triangles it's made of, we multiply that by 180°. Take your time to work that out. That comes out as 540. Now that is the sum of the angles in a pentagon. Now we normally want to work out the size of one of the angles. This particular question is a relatively tricky one because it's put this parallelogram inside the pentagon and we only actually want to know this little part of the angle here. However, we want to know what this full angle adds up to because then we'll be able to find any gaps in between if and when we can find the angles in the parallelogram. So there are five angles in a pentagon. So I would divide the total by five. 540. Five goes into five once. it doesn't go into four. So carry the four and then it goes into 48 times. So the total angle which I'm going to draw just here even though we don't need it there is 108° and we'll use that to work out that missing gap. So that part of the working regardless of the rest of the question and I have just gone straight into working out the interior angle of the pentagon for a reason because no matter on the complexity of the question I'm going to get quite a few marks just for doing that working out. So my formula is n minus 2 * 180. And if you're working out the interior angle, you also want to divide by the amount of sides. So it seems important to know that formula. n is the number of sides. There we go. And that is a really nice formula for it. So for the parallelogram, we need to know a few things about parallelograms. And the most important thing about a parallelogram when we're looking at angles is these opposite pairs of angles are equal. And the same with these two. And it's a four-sided shape. So they all add up to 360. That's one way of working this out. So we could work out the 117. Take those two away from 360 and then divide it by two to split it between the others. However, there is also a parallel line rule that you can apply in a parallelogram. And that's these angles here that make this Cshape. They are called co-terior angles and they both add up to 180. If I was to sort of draw this as parallel lines, there we go. These are called co-terior angles and they add up to 180°. So here we could just do 180 take away 117 and I'll work that out to the side. That's for this one here. Call it x 180 take away 117 and that is equal to 63°. So that one there is 63. We'll replace that. Put a little x here. That's 63°. So we're almost there. We found the interior angle of the pentagon. We found the interior angles within that parallelogram because this is 63 as well. To find this missing angle that the question's looking for, let's get rid of all these lines. To find this missing angle, we know the total is 108. And we know up to the edge of the parallelogram is 63. So to work out this final piece, we can do the total angle 108 take away the 63 and that leaves us with 45°. So there we go. Our final answer for this one is 45° and we have shown all of our working. Okay. So looking at surface area and volume of a cube. Now this question says the total surface area of the cube is 294 cm squared like at the volume of the cube. Now when we are looking at surface area it can help just to have a visual representation of a cube so we can actually imagine what this looks like. Now it says the surface area is 294 and that includes all of the surfaces of a cube. Now we might not be able to see a cube when it's drawn this way but if you think about a dice for example there are six surfaces to this cube. So in order to find the area of one surface and obviously this the thought process behind this is complex as it is because in order to find the volume of a cube we need to know the side lengths. Well if we know the area of one of the faces well then we can work out the side lengths as each one's a square. For example, if the area of the one face was 100, we would know that the side lengths were 10 because 10 * 10 is 100. And then we could work out the volume. So to work out the area of one face, we would need to divide by six. So if we do this to the side, 294 / 6. Six goes into 29 four times up to 24. And then five left over and nine times into 54. So the area of one face is 49 cm squared. Now if one of the faces has an area of 49 cm squared to work out the side lengths we're just trying to think well what number times by itself makes 49 and that of course is 7 cm. That is a square number. You can do the square root of 49 as well which is equal to 7. So now we know that each side length of the cube is seven. This is seven. This is seven. And the height is seven. To work out the volume of a cube, we just multiply those three sides together. So the volume will be 7 * 7 * 7. Want to take our time to work that out. But of course, we already know that 7 * 7 is 49. So we just need to do 49 * 7. We don't have a calculator for this. So we'll do this here. Make sure we get it correct. 7 * 9 is 63. 7 * 4 is 28 + 6 is 34. So 343 is our answer. This is of course a volume. So 343 cm cubed is the volume of this cube. Okay. So looking at the volume of a prism. Now we have mentioned working out the volume of a prism in a previous question. However, this one here has actually given us the volume. It says here the prism has a volume of 750 cm cubed. Look at the height of the prism. So the height is this part here. And it does tell us in the question the cross-section is a right angle triangle. We know to get the volume of a prism we do the area of the cross-section and times it by the length. So, if we've been given the volume, well, we could divide by the length to find the area of the cross-section. So, in this particular question, our first step would be to divide by 25. That's going to tell us what the area of the triangle is. That cross-sectional area, we need to take our time to work that out. It's actually not a bad one to do with bus stop division. 25 goes into 75 three times and then we have a zero. So our answer is 30. That is the area of the cross-section. So the area of the triangle now we know has an area of 30 cm squared. Now the base is five and we don't know the height. Now the formula for the area of a triangle is half base time height or base time height / 2. So what we need to do really here, we could put the numbers in and see if we can figure it out that way, but we know 30 is equal to 5* the height / 2. Now think about this logically and you can rearrange this as as a type of formula as well, but just thinking logically, if we're going to divide by two and need to get the answer 30, that means that we are looking for two numbers that times to make 60. What's the times to make 60? We then have it and get 30. So 60 has got to be equal to 5 * the height. And that's essentially just rearranging it anyway. It's tsing both sides by 2. So what number * 5 makes 60? Well, 60 / 5 is 12. So h has to equal 12 or 12 cm. So there we go. The height of the prism has to be 12 cm. Using a bit of reverse volume there, dividing the volume by the length that the cross-section goes through the shape and then applying a bit of area of a triangle to figure out that missing piece. Now, with exact values of trigonometry, there are only three major ones that I would be looking at trying to learn or remember. And if you don't know how to work out the exact values, there are two which I would really focus on. So this is one of them. Write down the value of sin 30. Sin 30, this is a really quick one for us to go through, is equal to 12. The other one is cos 60. And cos 60 is also equal to 1/2. So two really nice ones to remember. They are the kind of things I would have written down on a revision card. And then just before the exam, I'll be looking through my revision cards and just remember if I see sin 30 or cos 60 remember to write down a half. When we are using exact values or socket without a calculator, we might have something like this. So here is a right angle triangle says cos 60 is equal to 0.5 or you could leave that as 1/2 if you prefer. Work out the value of x. So we need to know you may or may not use formula triangles. If you do use formula triangles, then you will know that you need to label up the two sides in the question. So in this one here, we don't need the opposite side, but we do have the hypotenuse and the adjacent side. We just don't need to label the opposite cuz it's not been given to us and it's not one we're looking for. So with a and h, that is our cos formula. So C A H if you use a formula triangle to work out H we'd cover up the H and it's A / C the A is the adjacent side for C is cos with the angle so cos 60. Now this question has actually told us that cos 60 is equal to 0.5. So if I put that in place we have 4 / 0.5. Now, two ways that you can work this out. If you divide anything by 0.5, it just doubles the answer and you get eight. So, our answer is 8 cm, which makes sense if the relationship between the two sides is one is half the size of the other. The longer side would be the one that's the double double the length. You could also look at this in terms of a fraction over one. You could times the top and bottom by two, which would be 8 over one, which again is of course 8. So 8 cm is the answer. whichever method we use there. Okay, looking at a pictogram. So the key is obviously very important on a pictogram. This one does give us a key. It says that the rectangle there represents eight pictures. Question says the picture shows information about the number of pictures sold in an art shop in each of January, February, and March. Write down the number of pictures sold in January. Well, the first rectangle is 8, another eight, and another eight. So 86, 24, and our answer is 24. Nice and easy as long as we have our key. Says 12 pictures were sold in April to show this information on the pictogram. Well, we need at least one rectangle. Making sure that our rectangle looks about the same size as the one above. We don't want it to be misconstrued as a smaller version if we're not drawing it neatly. So that's eight of the pictures. We just need an additional four. So four would be half of this rectangle. Now if you have a look in the diagram they have drawn this one here which is a version of the four. So try and replicate any that they've done in the same way. So that's half there. It's kind of like a little C-shape and that represents four. And there we go. That would be 12 in total. Part C. What was the total number of pictures sold in these four months? So we need to add all of them together. So good job we've already written down those two. We have 8 16 24 plus a 4 that's 28. And then we have 8 16 20. In that one we just need to add all of those together using column addition. So 4 + 8 is 12 + the two is 14 and then 2 4 6 7 8. So 84 84 pictures were sold in total. And there we go. That's using a pictogram. Looking at some sampling and bias. Sampling questions are actually really nice. In here, we've been given a sample of what they've told us is 30 students. Each students chooses one place and the table shows the information about the results. So, here it says how many of the 195 students you think will want to go to the theme park. Now, this is all just about proportion. If we look that there are 10 students in our sample of 30 that want to go to the theme park. And if we bring that down and write that out, that's 10 out of 30. Now that particular fraction there is a relatively nice fraction. We can simplify that. They both divide by 10. And if we do divided by 10, we get 1 over three. So we know that 1/3 of the students want to go to the theme park. Now in this question we have 195 students going on this trip. So we just need to know what 1/3 of 195. Now to do that we'll use bus stop division. So 195 / 3 3 goes into 19 six times up to 18 remainder one and then five times into 15. So our final answer we would say for this one 65 students and there we go there is our final answer for that one. For part B it says state any assumptions you made and explain how this may affect your answer. Now when we do this we are assuming that the students in the sample represents the rest of the students that are going on the trip. So we are assuming that the sample represents the rest of the students. Now it says here how might it beffect how might it affect your answer if our assumptions essentially are not correct. So if we are assuming that the sample represents the rest of the students of course there is a scenario where it doesn't. It might be that they are the only 10 people that want to go to the theme park. And then of course our answer would be a lot lower. It might be that everybody else outside of those 30 wants to go to the theme park. The two extremes there, but that would mean a lot more would want to go to the theme park. So if not, our answer may be higher or lower or maybe more or less. I'll just write more or less. You could write this in your own words, but the really important part there is that word where we said represents. The sample represents the rest of the students. If not, if not, our answer may be more or less. There we go. That's sampling and bias. Onto a stem and leaf diagram. We are finding the median and the range from a stem and leaf. So, it says here the stem and leaf diagram gives the information about ages of people in a club. And again, the key is really important. tells us that 4/2 is 42 years. Now to find the median, we want to know where the middle number is. And to do that, we want to know how many are in the stem and leaf. So if we go along counting them, we have 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20. So there are 20 people. Now, to find the middle, particularly when there's an even number, we need to be really careful. Now, you can apply a logical approach to find the middle, or you can just cross numbers off either side. Most people just like to cross the numbers off. I like to underline them. So, I'm going to do 1 2 3 4 5 6 7 8. I know I can do up to eight because if I do another eight, that's only at 16 and not 20. One, two from the back this time. Three, four, five, six, seven, eight. Once you've done this one and you're getting close to the middle, just slow down a little bit. Make sure you don't start crossing off extras and miss the middle. So, one more from the start and one more from the end. And then we actually have two numbers in the middle. So, when this happens, when there's an even amount of people, it will happen. You won't get one in the middle. We want to find what's in the middle of those two. So, we have 51 and 57. To find what's in the middle of two numbers, you can add them together and divide it by two. Or you could write all the numbers out from 51 to 57 and again cross off until you get to the middle of those. But adding those two together gives us 108 / 2, which is 54. And you might have known that four was in between one and seven, and that's fine as well. So there we go. Our median is 54. The range is really nice for us to find. Here's the biggest number, which is 74. And here's the smallest number, which is 31. Just making sure you're using the key. The range is the distance between the biggest and smallest. So 74 take away 31 would be 43. And there we go. That would be our range. Okay. Probability from a table. So when we are looking at probability within a table, we need to know that all of these add up to one. As long as we know that, the rest is okay. This one does have a nice aspect to it, but we are missing two. But it says the probability of red and yellow is the same. Complete the table. So really the first step is to write down what's left. So 1 take away the 0.2 that's in the table leaves us with 0.8. We could write that as 0.80 as it might make it easier in a second to divide that by two. So these two have to add up to 0.8. Now, that's a nice number to have to divide by two as they're both the same as they're both just going to be 0.4. However, just remember if it was something like 0.7, you might want to write that as 0.70 so it was easier to divide by two. But there we go. 0.4 for both of those. It now tells us that there are 12 blue counters in the box. Work out the total number of cubes in the box. Well, if we know that 0.2 two here is 12 counters. This is a really nice one because 0.4 is just double that. So if 0.2 is 12 counters, we'll just write that here. Then next to that, which is double, we would times that by two and that would be 24 counters. And yellow is exactly the same. So that would be 24 counters as well. So actually this is a really nice one. would be more difficult if one of them was something like 0.3. But potentially not that bad if we are finding the total anyway because if 20% of them are blue, not forgetting these can also be written as percentages. These decimals 20% and 40% there. If 20% is equal to 12, then we could actually instead of adding those all up, we could just say 20% equals 12 and use that reverse percentage approach to go straight to 100% by tsing by 5, which is 60 counters. So that's actually probably the quicker method there. But otherwise, we could label it on the box or on the table and then add them all together. 12 + 24 + 24. There we go. Final answer 60. And looking at this one, we have a frequency tree. So 72 people each took a driving test. 20 of the 32 adults who did the test passed. So it tells us 72. So we can put that into our frequency tree straight away. 20 of the 32 adults. So it says there are 32 adults. That part just there. And it says 20 of them passed. So 20 passed. And straight away that means we can actually work out that 12 failed. It says six of the children who did the test failed. So that we can put in straight away. Six of the children failed. So children that failed is down here. Six. Use this information to complete the frequency tree. Well, we can work out the rest now. 72 took the test. 32 of them were adults. So 72 take away 32 leaves us with 40. So 40 children and six of them failed. So for this box here we have 40 children. Take away the six that failed and that leaves us with 34 that passed. That's the frequency tree completed. It says one of these people is chosen at random. Work out the probability that this person is an adult who failed the test. Now if we find that part the adults that failed is this box just here. So there are 12 adults that failed. One of the people is chosen doesn't specify adults or children. So it's out of 72. So there were 12 adults that failed out of 72 people. And I don't need to simplify that at all. If it changed the language there though and it said one of the adults is chosen, work out the probability this adult failed. Well, it' be 12 over 32. But it doesn't specify that we're picking from the adults or the children. Just says one of these people is chosen at random. So that would be our probability. And onto a vin diagram where we also need to put all of our numbers into the ven diagram. And we don't have any set theory in this one. We will discuss that. It says the even numbers less than 19 are all the numbers that need to go somewhere. It then tells us the numbers that go in A and B. And first we want to see if there's anything in both. Check that the ven diagram is labeled. It is labeled A and B. So we have a six in both. And we have an 18 in both. So six and 18 go into the middle or into the intersection. We then have 12 the only number left in A and 2 and 14 the only numbers left in B. We just need to put the rest of the even numbers in less than 19. We have two. So, we need four, we've got six, we need eight, we need 10, we've got 12, 14, we need 16, and we've got 18. So, there's numbers on the outside. And that's the ven diagram completed. Now it may ask you things like what numbers are in A union B or A intersection B. So if it asks you about these, remember the union is anything within all of the circles including the intersection. So if it asks you about the union, that's the part of the ven diagram you would be looking at. And if it asks you about the intersection in that scenario, you are just looking at that intersection between the circles. So here the numbers in A and I'll refer to them as U and N. In A N B you've got 6 and 18. In AU B you have 68 plus the 2, the 12, the 14. So it might ask you probabilities. It might ask you something like what's the probability a number is in A and B? Well, there are two numbers in the intersection and overall there are nine numbers in the ven diagram. So, it's two out of nine. There's lots of different questions you could be asked, but this one here just completing the ven diagram. And onto a probability tree. Alice has two bags. Alice takes at random a ball from the first bag, which we can see on the probability tree. Three out of 10. For red, we don't know green. She then takes at random a ball from the second bag, which we can see five over 9. red, we don't know green. Complete the probability tree diagram. So each pair of branches have to add up to one. In the case of fractions, they means the numerators have to add up to that denominator. So with three out of 10, the other one has to be seven out of 10. And that's that pair of branches done. They don't link to anything else. I'm now going to look at the other pairs of branches. So look at this one that has a denominator of nine and red is five out of nine. So they have to add up to nine. So that would be four out of nine. Those two fractions now add up to one. Now in this question it doesn't say anything about the second bag being different depending on the red or the first or anything like that. So these fractions are all just the same. 5 over 9 4 over 9. That pair of branches is the same as the pair above. So there we go. That's our probability tree completed. It now says work out the probability that Alice takes two red balls. So if we are going for a red and then another red, that's the route that we go down. Now all we do is identify the fractions on that route and we're just going to multiply them together. Write the working out down. So 3 over 10 * 5 out of 9. 3 * 5 is 15. 10 * 9 is 90. And you do not need to simplify your answer. That right there is our probability that Alice would take two red balls. And there we go. That's probability trees. [Music]