Top 5 A Level Maths Topics You Should Master Before September Starts

So you guys are really tenacious because even though I've said that you shouldn't really do much during the summer, a lot of people have been saying that they're starting to revise, they're starting to study, what topics should they study etc. So what I've done is I've put together the top five topics that I would like all of my students in year 12 at the beginning of the year to be able to do, whether they're A-Level Math students or A-Level Further Math students, and I'm also going to tell you how I would actually study them myself. Now the really good thing about these topics is they are universal, they apply to Edexcel, AQA, OCR, CIE, whatever example you're doing because it's really fundamental maths, and they are all common for GCSE, we're just stretching it a little bit. I would not recommend learning something that is completely new like exponentials and logarithms, mechanics, binomial distribution, differentiation and integration because you need a teacher to explain them to you to make your life just easier and make it more efficient. So let's just jump straight in with the first topic which is just generic algebraic manipulation which is chapter one in the textbook and if you look all of this stuff is something that you do in GCSE Maths and I've given you an example GCSE Maths question of the kind of difficulty that you should really be aiming to master by the time you're in September or just in September in general. The reason why this is so important is because it's not because there's a lot of questions with indices as the question itself, so something like this, this does appear but it's far more common that in almost every question that you do there's going to be indices involved and so you get like one mark for rearranging this indice thing, if you get that wrong you just get this cascading error where you get, you know, you lose mark for the indice and you lose the accuracy mark and you lose a method mark and it's just a bit tragic. A lot of the other topics that we cover in A-Level Maths also use indices as part of them such as integration, differentiation, exponentials and logarithms and so being, when I say mastering this I'm saying make zero mistakes, it is really really important. I should also note that in this first chapter as well there is a couple of new things like disguised quadratics which leads me on to the next part which is quadratic equations. So this is the second most important topic that you should be really really good at, again I've picked a GCSE question to give you some context, you should be able to find turning points from quadratic graphs using algebra, but again if you look at this chapter it's very very similar, so solving equations for pink and square, this is something that occurs all of the time in A-Level Maths, A-Level Further Maths and even beyond as well, quadratics are super super common. Now it is true that the A-Level Maths calculator can solve quadratic equations for you but it's also true that most questions in A-Level Maths will say solutions based entirely on a calculator will not be accepted, which means if you don't show how you solve the quadratic equation they don't give you any marks whatsoever. Plane to square, similar, functions is something that a lot of students struggle with in GCSE, quadratic graphs again you've seen that before, but then the discriminant is something that is new and the reason why I put this in here as well is that you can get kind of a taste of the new stuff in A-Level Maths, it's not too tricky but the discriminant is used everywhere, it's used so much and you'll be able to get an idea of what A-Level Maths is like where you learn something and then you apply it in like 500 different contexts, which in my opinion makes maths far more interesting because you see this tool that you have and you look at all the different ways that you can use that tool. Modelling with quadratics, I think it's a good point of exposure because modelling is incredibly, incredibly, incredibly common and important in A-Level Maths, not just in pure maths but in stats and mechanics as well, these questions come up all the time, they're worth a significant amount of marks and you have to do it for all kinds of different things, so starting with quadratics, something familiar makes a lot of sense to me, so that's why quadratic equations are number two. For number three, coordinate geometry but hard. Now this is a really important topic because A, it comes up all the time in A-Level Maths, Year 12 and Year 13, but it also comes up in Further Maths, so in Chapter 2 of Further Maths you do something called Argon Diagrams and it's basically, instead of using xy coordinates, you use imaginary real numbers but you use the exact same techniques as you would in regular maths, so this coordinate geometry is really valuable. For those of you that want to kind of really get into those top universities, you're probably going to be doing things like maths challenges and maths olympiads, coordinate geometry comes up a lot there. If you have to do the TMUA for a lot of different degrees or you have to do the STEM, coordinate geometry comes up a lot. Interview questions, coordinate geometry comes up a lot because it really tests your problem-solving skills and your maths skills, so this is one of the most important things to get really really good at, but again I put it third because you can't really do it properly without the first two things. And by the way, these chapters here are from the actual Pure Maths textbook, I will put the link in the description, although it's in my other videos as well, and again you've probably seen all this before, why is it so explicit, common, boring, but again we're looking at the problem-solving and the coordinate geometry itself. So while these chapters sound easy, they actually aren't and there are a lot of students that struggle with questions like that in year 12 and year 13 and also in their interview prep and their university entrance exams. And once again we have some modelling because again modelling comes up all the time, lengths and areas, very interesting stuff and you'll use so much from GCSE, so there might be a time where you use the sine rule and the cosine rule, there might be a time where you use all these different ratios and stuff that you've seen before. With circles, again if you didn't do GCSE further maths, you do need to learn one thing as well, which is the equation of a circle is actually slightly different, because in GCSE, regular maths anyway, the equation of the circle is always centred at the origin and now we can put the circle anywhere on the coordinate axis, so it changes the equation slightly. But again, super super important, circles and triangles, tangent and chords, again if you're looking at maths challenges, maths olympiads, entry exams, all that stuff, this is just bread and butter stuff, it comes up all of the time, and so that's why I put it at number three. For number four, trigonometry. Now it's not just SOHCAHTOA, it's not just the sine and cosine rule, although that's here as well, but I want to make this really really clear, trigonometry is again, after maybe modelling and I guess algebra in general, trigonometry is the biggest part of A level maths. I'm not just talking about year 12, I'm talking about year 13 as well, and I'm also talking about mechanics. It comes up so much in mechanics, where you use it all of the time, all of the vector stuff we do in mechanics, all the vector stuff we do in regular maths, trigonometry, it just is everywhere, everywhere, and so getting really good at it is so vital. Now of course, starting off with stuff you already know, cosine rule, sine rule, area of a triangle, solving triangle problems is where it gets a bit more interesting, where we use things like bearings, which again comes up a lot in A level. The graphs are also vital. Now when we get on to chapter 10, where you start working on multiple values, if you remember those questions back in the exam, I think it came up in 2025 actually, where they gave you a sine graph and they said, you know, sine 30 equals a half, what other value gives you a half, right, and it'd be like sine 150 for example, 120, whatever. That kind of thing, we do a lot, and I would thoroughly recommend that you use the graphs instead of the something like the Karst diagram that is in the textbook, but I'm going to leave it up to you. Personally, I think the graphs are far more simple, and it's also just more mathematical than the Karst diagram, but whatever, you do you. Again, there are a lot of people online that do videos on this kind of stuff, people like Bison Maths or Zeeshan Maths, that'll help explain it and they're very good, but if you can master these two chapters, this will take you a very long time. For context, chapter 10, I spent four double lessons on it at least, so just again for context, a double lesson is 100 minutes, so I spent around 400 minutes doing this. Yeah, right, it's a really big thing, take your time, this will take a lot of it, but again, it's so vital for years 12 and 13, to gain that head start would be quite nice, and last but not least, was the fifth one, rearranging. I know, that seems like a bit of a step down from all exciting stuff we've been talking about, but some of the hardest questions in year 13 maths are show that questions, where they'll say show that this can be written as this, or show that after I differentiate that, I can write it like this. Yeah, happens all the time, and not just that, but when you're using equations and formulas, you're going to be rearranging them, right? Sure, cosine rule, can you rearrange for the angle? I don't know, can you? Why, because mx plus c, can you rearrange for the gradient? All these kinds of things, it comes up all of the time, so again, it's not just, if I give you a rearranging question, can you do it? It's, if rearranging is part of a question, where you might only get one mark, or even no marks for it, can you do it quickly, because again, it's not worth any marks, and efficiently, like effectively, can you actually get the right answer 99.999% of times? This is something that a lot of GCSE students tend to be weaker on, and when I'm marking those year 12 assessments that we do throughout the year, rearranging mistakes happen all the time, especially with further maths students. If you're doing A-level further maths, the equations we use are quite complicated, and so being able to rearrange complex things is very important. It's really important for A-level maths, year 13 maths especially, both A-level further maths, you're going to be rearranging stuff, and honestly, if you're really good at maths, but you're rearranging is letting you down, you're just leaving marks on the table for no reason. So, let's move very quickly on to how I study this. First things first, is the textbook genuinely is quite good. It comes up with one or two or three examples, and then it gives you some questions to do, so just working through the textbook is very good, but as I say with anything when you're revising, it's not just, can you do it? It's, do you understand what you're doing? I say this all the time, because if you understand it, you'll learn it a lot faster than if you're just doing the questions. If I gave you 100 quadratics to factorise, okay, sure, you might be able to do it, but you're going to forget it, unless you understand why you can factorise in that way. So, although that might sound silly, even with stuff like this, especially with the discriminant, this will be the first time where you're going to see, okay, do I understand what this actually means? Do I understand why it works? Do I understand what's happening? So, always think about the understanding. So, what happens if you don't understand? A few options. One is AI. AI has gone to the point where it can explain things to you, so why not try that? Two is YouTube. So, again, these are the three options where you can, you know, look at Bison Maths, Eshan Maths. I have some videos on a lot of the topics as well, if you want to look at those. And the third one is, of course, starting tuition early. Honestly, I'd recommend starting tuition from September, not from August, but that way, in September, you can just say to your tutor, okay, look, I want to go through this because I didn't understand it when I tried learning it. And doing all the questions. Now, just so you know, there are mixed exercises. This is the most important thing to be doing, because these parts here, so all these little chapters have their own end of chapter questions, right? So, you have questions just on this, questions just on this, questions just on this. The mixed exercise is questions on the whole topic, and it won't tell you what's what. So, that tests your actual understanding, which, again, going back to the discriminant, that's going to be very interesting if you can use the discriminant when it's not obvious you can use the discriminant. That's how you test your understanding. There's also review exercises, where they just mix up questions from all five of the chapters. Good luck with that. And that's the point, right? Testing your understanding. But again, you only have six weeks off, right? Well, seven more weeks as of this video, anyway. Take your time, learn as much as you can, if that's what you want to do. It's not compulsory, but I thought I'd make this video just in case you really, really were desperate to start 11 months.