Algebra is a method of solving number problems that have a number missing in the calculation.
For example
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For simple number problems we can just use number knowledge and/or number logic.
An excellent place to start is with the solve me mobile puzzles: SolveMeMobiles
Numbers can be imagined to be heavy and need to balance each other. On these problems, we figure out an unknown number that will keep the mobile in balance.
The job of algebra is to give us a way of writing or talking about unknowns while they are unknown.
In the image above, the unknown ‘weight’ is the moon shape. We could say, ‘how heavy is the moon shape when the mobile is balanced?’
You can figure it out with a little bit of multiply, dividing, adding, subtracting – the usual stuff.
That’s exactly what we do in algebra. Its just, we write the problem down in a lines of writing using numbers and letters which can be much more confusing than the nicely drawn diagram above. In fact, to transcribe the diagram above we might write,
Find 
when 
given that 
and 
.
Wa-at?
In an equation we are usually trying to figure out how much an unknown is worth. At other times letters appear in formulas and can have different values – these are called variables. Either way, the letters represent numbers, and can be treated as if they were numbers – we can multiply them, divide them, add to them, subtract from them and so on.
Sometimes we’re taught how to perform several steps at once (eg, ‘cross multiplication’, or ‘FOIL’). It’s quite easy to get lost in the world of algebra. It takes time to make connections between ideas. The more you go over the ideas the more the ideas fall into place with each other (especially if you are in the fortunate position of needing to explain an idea to someone else).
The word algebra comes from a book written by Al-Khwarizmi.
Read more about Al-Khwarizmi here: St Andrews University History of Math Website.
There have been lots of different ways to write about unknowns in the long history of math, however we’ve settled for a language that uses letters for unknown or variable quantities in equations, which behave just as if they were in fact visible to us as numbers.
For example:
means 
. Using the distributive law, it means 
Now consider 
. We can’t evaluate it unless we somehow know the value of 
. Even without knowing, we can still use the distributive law and say 
.
In other words, when we have letters instead of numbers we have some limitations, but we can keep going as if they were in fact known numbers by treating them with all the same rules.
Here are a couple of things that might not be obvious, and can cause confusion from day one:
means four, plus 
times whatever number represents. When we put a number and a letter together, we mean that they are multiplied together. The rules of BEDMAS tell us that 5 multiplies , it doesn’t add to 4. Think of as 
. Not as 9x.
means four, plus 
. When we don’t put a number in front of a letter, it means that there is one of them!
The math language we use tries to achieve balance between clarity (having people understand it) and efficiency (writing as little as possible). This mathantics algebra video explains the language really well. https://mathantics.com/lesson/what-is-algebra