## Equations and Graphs

A really useful skill to develop when learning mathematics is graph sketching; although some of the GCSE and A Level textbooks do have a short chapter covering this somewhere amongst the pages, unfortunately the value of graph sketching is, for the most part, almost invariably overlooked. Students think that this is something that they only need to do when they are asked to do it and even then it’s clear in many cases that they don’t really understand what it is that they are doing.

An equation such as $y=x^2-5x+7$ gives a clear, unambiguous, relationship between $x$ and $y$. So what? Well this means that for each $x$ value that I put into the right hand side of the equation I will get a corresponding y value. So for example if I choose $x=3$ (there is nothing special about this choice of value for x; I could just as easily have chosen $x=5.87$ or $x=-29$), I simply replace any $x$’s on the right hand side with $3$ to get $y=3^2-5×3+7$ and so $y=9-15+7$ and therefore $y=1$.

For many GCSE and A Level students this is fairly straightforward – but they miss the important part. This $x$ value and its corresponding $y$ value form a pair of coordinates, in this case $(3,1)$ – remember that the $x$ coordinate is the first number, $3$, and the $y$ coordinate is the second number, $1$.

Again – SO WHAT?

Well if I do this for several x values then I get a few more corresponding y values and so more pairs of coordinates which I can start to plot on some $xy$-axes. The more $xy$ pairs that I find the more points I can plot; what emerges is the graph of this equation. In many cases you don’t even need to plot a great deal of points to figure out what the shape of the graph is going to look like; many equations can be grouped together into families of equations – equations that have certain similarities – and the graphs of these families of equations (quadratic, exponential, logarithmic) have characteristic shapes and behaviours.

This is important – the equation and the graph represent the same thing but in different ways; the equation determines what the graph will look like and the graph will (at least in theory) determine the equation. If I choose any point on the graph and study the coordinates of that point I will find that the relationship between the $x$ and $y$ coordinates, for all points, is the relationship described by the equation and, moreover, any $x$ and $y$ values that are related via the equation WILL, without exception, be the coordinates of a point on the graph of that equation. So now, rather than seeing an equation as a string of miscellaneous symbols on a page I can start to draw (in some cases of my own volition) the graph of an equation and from this I can see (in the literal sense) the behaviour of the equation at a glance. The equation suddenly becomes more than an equation – it has more character to it. Drawing the graph of an equation might seem like more work – why would you inflict that on yourself? But that isn’t the case. By knowing what the graph of an equation looks like you get a visualisation of the situation and this usually means that overall you have don’t have to work anywhere near as hard to get your solutions.

Sadly, many GCSE and A Level maths students never get to grips with this. Not because they CAN’T draw graphs but because it is never made clear to them that this relationship between equation and graph exists. They go on thinking that in order to draw the graph of an equation they have to have been told beforehand what the graph of that particular equation looks like. This is a disaster for them! I often ask my students to sketch the graph of an equation and to start with they may look at me blankly or say, “I’ve never been told what that looks like”. My response might usually be something like, “Well think about what the equation is telling you is going to happen for different values of $x$”. This might not hit home straight away but eventually it does in most cases and, finally, they understand that the graph (even of unfamiliar equations) can be figured out themselves by thinking about what the equation is doing at various (well chosen) values of $x$.

Once you get your head around this then you’ll find that maths becomes ten times easier.

## The fear of failure

I just want to talk about something that really holds a lot of people back when it comes to learning maths; something that stops (too) many people from achieving what they can truly achieve – the fear of failure.

The fear of failure and of making mistakes is a very common issue amongst maths students (though, I’m certain that it’s also the case in other subjects as well). Failure (or at least what is perceived to be failure in someones own mind) and mistakes are always seen as a bad thing. Sometimes failures and mistakes are bad things because the consequences may be very severe – but when you’re learning GCSE or A Level maths, the good thing is that no-one dies if you fail or make a mistake; there is no catastrophic nuclear accident; the world continues to turn and the sun continues to rise and set.

Mistakes are a necessary part of learning and doing mathematics at all levels; and the possibility that you might fail to solve a problem is something that you have to learn to accept as this will ALWAYS be the case even if you happen to be called Albert Einstein (OK, he wasn’t a mathematician but you get my drift).

I know that it feels good when you get the correct answer to something and you succeed in writing out a correct solution – but if you only continue to do what you know how to do then you will only ever be able to do what you know how to do; in other words you will stop learning. The possibility of failure is something that I very often have to work very hard to get my students to accept. Sometimes, and this is particularly true of people who have done well with GCSE maths and moved on to A Level maths, students are so used to getting things right first time and without any difficulties, once they’re presented with an actual problem that they have to solve they will give up. The thing is, it’s not that they can’t solve the problem because in many cases they can, but just that they think that if they make a mistake that they have failed miserably and that only ridicule and embarrassment will follow. They are so afraid of making a mistake that they won’t even make an attempt – they just say, “I haven’t been shown how to do that”, or simply “I don’t know what to do”.

As a mathematician (or student of mathematics) you have to be willing to make an attempt at applying your knowledge and accept that you may (and in all likelihood, will) make mistakes; you may even be unsuccessful in finding a solution for some time and you may NEVER find a solution. Have you failed in these cases? No – not at all. You have failed if you make no attempt at a solution; you have failed if you’re not prepared to make the mistakes that you need to make in order to learn.

Sometimes I have to put my students in situations where they HAVE to make an attempt at solving an unfamiliar problem and where they might make many mistakes along the way; I will not give any clues about what to do until a good attempt has been made by themselves to find a solution. Cruel of me? Not really as this is what they’ll have to do in the A Level or new GCSE maths exams so it would be cruel and wrong of me to not do this and only have them do things that they can easily do. By doing this I’m not saying that I expect them to find a full solution and I even say to them that I’m not bothered if they don’t find a solution; what I do expect is that they put themselves out there and try things out; I AM bothered if they don’t make any attempt even if they claim that they don’t have any ideas. I have to get them to understand that mistakes are a natural part of learning and doing mathematics; that no-one is going to laugh at them if they make mistakes; that their otherwise impeccable reputation will remain intact despite their mistakes; and that they will learn more from making these mistakes than by not making them.

I can see their faces contorting when they first write something that they know might be wrong – but this is what doing mathematics is about; trying things out until you find something that works. As a mathematician you’re not expected to be able to blurt out the correct answer or solution immediately upon being presented with a problem. What you are expected to do is to work to find a solution and this means making umpteen false-starts in the process. Once my students accept this fact and free themselves from the fear of failure (whether real or perceived) then I know that they’ve taken a huge step in the right direction.

## Common A Level Maths mistakes

Here’s a nice long video for you – I realised that the A Level exams are just around the corner for 2017 and the panic-frenzy will be really kicking in soon. But ther’s still time to get well prepared for your exams if you start your revision and everything now.

This is a video that I made to point out some of the bear-traps that people very commonly fall into when studing for their A Level Maths and Further Maths and when preparing for their exams. All of the points that I make in this video are based on my personal experiences as a maths tutor over the last several years – they are mistakes that I see people make time and time again, year in and year out and they are mistakes that could cost you quite dearly if you persist with them either knowingly or unknowingly.

Obviously I can’t cover every single eventuality but I’ve tried to focus on the main things that people do wrong. Equally obviously, there is no magic wand that I can wave to make everything better and to guarantee the result that you want. Whether you get the result that you want is entirely down to your own level of work and your own attitude but I hope that the points that I make in this video will point you in the right direction at least.

If this video was useful to you then you might also want to watch my other A Level Maths videos – one is for A Level Maths and Further Maths and the other video is for those taking the STEP Papers.