Today, we're going to briefly talk about tackling some of the problems you will encounter while taking your GCSE maths exam - we're going to start off pretty simple, but before long we will be using basic maths skills like arithmetic and our fractions knowledge, to solve trickier problems: equations, geometry problems, so on and so forth!
While we're talking about maths, let's consider a probability question: many learners don't like maths, its got a bad reputation as being difficult to understand, and students and adults alike are apprehensive about attempting to solve maths problems, particularly in trickier domains like calculus and trigonometry.
Given that so many people don't like it, it's probable you might be in that group. If so, we're hoping to change that, or at least make you a little more comfortable, today!
However, today we're going to talk about how adopting a logical approach to solving mathematical problems allows us to develop our scientific minds, and with some practice: solve problems from memory, instead of referring to textbooks!
The trick to learning how to succeed in maths, is practicing all the skills you learn, from primary school through to your GCSEs, A levels and beyond, so that you know how to break down problems into smaller questions that you can answer, until you've found an overall solution.
The key thing to remember is this: however difficult your problem is: whether its an algebra question, a question about inequalities, fractions or coordinates: each "type of problem" has the same steps to solve it. You just need to develop your reasoning skills, so that you know which set of steps to take.
This means that all fractions problems have a set of steps you can work through to solve them, and all algebra questions have a set of steps to work through. You'll get to the answer eventually!
One final bit of general advice: if you know of a gap in your maths knowledge, work to fill it ASAP! Mastery of a topic that you were unsure of to start with is a surefire way to improve your maths knowledge, and therefore your grade!
Don't forget: the maths skills you pick up now will serve you well for the rest of your life: regardless of your career choice: engineer or retail worker - a large proportion of careers will mean you end up using some maths! A math tutor may seem unnecessary but if you're struggling, they make make you maths learning a lot simpler.
Advice for Solving Common Problems
Just like we said earlier, the basic method to learn maths needed to solve a type of problem will become the foundation of your maths knowledge, and will be key when trying to develop your problem solving skills, later!
Really, you can think of it like this: from primary school through to your GCSE maths revision and A level exams: maths problems are really just testing how well you can learn things off by heart.
That's all you need to be able to do to succeed at maths, regardless of whether we're talking about algebra, geometry, or some other class of problem!
Speaking of algebra and geometry: make sure you have a good handle on these topics, as many of the problems you'll face during your academic career (including during exams!) will focus on one of those areas (they might even overlap!)
A great way to start solving maths problems is to imagine them like puzzles!
You have a series of clues set out for you in the question, and it's your task to piece these clues together: this will lead to you performing a few calculations, and with any luck, you'll have the answer to the question posed to you! In primary school, you were likely introduced to "word problems", and really, a lot of GCSE and A level maths exam questions are just "word problems".
A key part of being able to solve them is being able to extract the information you need from the question.
The first thing you will need to make sure of if you want to improve your maths skills, is make sure you are familiar with all the content in the lessons you've had with a teacher! Just looking over old maths problems is pretty dull, so see if you can make up some problems yourself that use the same techniques taught in class.
Really, your maths teacher is telling you the key principles you need to solve a maths problem of a particular type, and your job in the exam is to put these principles into action!
So, to really drive this home, during your maths revision focus on regularly attempting problems of different types, and if you get stuck, look at the notes you made during your lessons.
Do not just read over principles: practice applying them!
Search around for online tuition for maths and math worksheets, and if you have a textbook, consider redoing some homework exercises. There are plenty of free maths resources around, just keep your eyes peeled.
So now you've mastered the basic principles behind solving different types of problems, here are some essential things to remember to make sure you best use those skills to solve the problem put in front of you:
- Make sure you have a calculator to hand, if you're allowed one!
- Take time to read the problem, several times if you need to! It contains all the information you need to answer the question, and if you only glance over it, you're likely to make an easily avoidable mistake!
- While reading the problem, underline or otherwise highlight any clues that jump out to you. These key bits of information will be your ticket to the answer.
- If you're unsure of how to start the problem, try several different theorems that you know and see if any of them look promising. You can cross out any rough work, nobody will mark you down for it!
- Check your answer, if you can! As you will see later, there's usually a way to make sure the answer you have fits.
You are now well on your way to becoming a mathematician! Nevertheless, you're still likely to run into difficulties with trickier problems, and it's important you don't become discouraged if you do!
Practice makes perfect!
Concepts for Solving Equations
Before we start solving equations, let's just make sure we are all clear on what an equation actually is, because this is one of the most common algebraic problems you'll come across, and being able to solve them is crucial for any mathematician. So, in short:
An equation is an equality which consists of at least one variable (usually called 'x') - you must find the value of any variables in order to solve the problem.
Way back in primary school, you would have learnt basic addition and subtraction, and it looked something like this:
Hopefully, we're all following so far. And if I wrote something like the following, hopefully everybody would still be keeping up:
This sum works, because both sides will become equal to eight once you add the terms (a fancy word for numbers) on both sides up.
Notice I said "sum" there? What you're really looking at is an equation, I just haven't put any variables in it yet. Let's see what happens when I do:
Now, this is the same as the sum (or... equation) that we started with, I have just replaced 6 with "x". Hopefully then, it's not too large a jump to say that x is 6, and thus you find 8 on the other side.
If you aren't too clear on how I reached that conclusion, there's no need to panic - I'm going to go into more detail in a moment.
The equation above is likely easier than something you would encounter once you get to GCSE level - it's more like a year 7 or year 8 equation; but just like before: every equation has the same steps to solve it, and once we've mastered these basic principles, we're well on our way to solving complex problems!
From now on, you'll see equations everywhere in maths. Especially if you choose to pursue a degree in a STEM field! It's probably best to learn how to solve them, then...
Solving Linear Equations
The kind of equation we just saw is what's known as a linear (sometimes called a 1st degree) equation, and it's called this because there are no powers in the equation.
They're the easiest type of equation to solve, but also require all the fundamental skills you'll need to solve trickier equations, later.
So, here's a quick step-by-step rundown of how to solve linear equations (sometimes called linear functions), and then we'll see an example:
- Isolate 'x'. This is the variable you're trying to find, so you want your equation to be in the form of "x = ...".
- To isolate 'x', put as many things as possible on the other side of the equation. If two things are multiplied together, you can't only move one of them across.
- You can do any mathematical operation you like on an equation, provided you do the same to the other side (for example: if you multiply one side of the equation by 3, you must do the same to the other side).
- Carry out any mathematical operations necessary to isolate 'x' completely, which should leave you with an equation in the form "x = ...".
This all sounds a bit complicated when written out like that, but let's see an example which should clear things up for you. The following example is GCSE level, but on the easier end:
Now, let's see how to solve it:
I start off by adding 5 and 2 together, and then 3x and x. I've done this to try and get as close to isolating 'x' as possible. I'm not quite there yet, but this is starting to look like the simpler equation we saw before.
Now, remember I said that we can perform any operation we like on an equation? We're aiming to get 'x' on its own, and to do that we need to get rid of the 4 on the left hand side of the equation.
The left hand side of the equation is really saying "4 multiplied by x", so to get rid of it, we're going to do the opposite: divide. And remember: if we do something to one side of the equation, we have to do it to the other side, too:
Now, on the left hand side of this equation, I'm multiplying by 4, and then immediately dividing by 4 - so really, I'm not doing anything at all, as these are opposites.
This means we can cancel the 4 out of the left hand side altogether. However, on the right hand side, 7 and 4 are not opposites, so we can't do anything with them, or simplify it any further.
This leaves us with our final answer:
And there you have it! By following the principles we set out above: trying to isolate 'x', and performing operations to both sides, we've found our answer!
How do we know it's right though? It's simple, really: take out your calculator, and put in our original equation, but this time, every time you see 'x', put the number we just found to be 'x'. If our calculations were correct, then you should get the same answer as you'll find on the right hand side of the equation (in this case: 7).
Of course, another way to improve your mathematics understanding is to find a good maths teacher, or a one-to-one maths tutor. Perhaps SuperProf could give you maths help?