Imagine that you wanted to calculate $3+4\cdot2$. The most logical thing might be to read from left to right, just like you do when you read a book. Let's try it! The first number is just a three: $$3$$Then we have plus four:$$3 + 4 = 7$$And finally, we have that result times two:$$7\cdot2 = 14$$So it seems that $3+4\cdot2 = 14$. But this is not the only way of doing it. We could read from right to left instead. After all, some languages are written that way. In that case the first number in $3+4\cdot2$ is a two: $$2$$And then we multiply by four:$$4\cdot2 = 8$$And finally, the result plus three:$$8+3 = 11$$So $3+4\cdot2 = 11$. But that can't be right! Before we had $3+4\cdot2 = 14$! It can't be equal to both $11$ and $14$ at the same time. To fix this, we have to decide which method we mean when we write things like $3+4\cdot2$. Moreover, everyone has to use the same method. Imagine how confusing it would be if people did this differently. They would be speaking completely different languages.

Before we move on to the rule for calculation I want to explain why we get different answers to begin with. (Skip this part if it doesn't make sense).

I shall try to illustrate the two different ways of calculating $2+1\cdot3$. 

Left to right. First we take two plus one: 

Then all that times three:

But in ``right to left", $2+1\cdot3$ is one times three:

And then plus two:

You could say that in ``right to left" the $2$ does not get the effect of the multiplication.

The rule is actually neither of those I suggested above (left to right and right to left). The rule is like this: multiplication and division goes first. I'll explain what I mean using an example.

Say that I wanted to calculate $4\cdot4 + 3\cdot2$. Here numbers are being multiplied in two places: 

I have to do these multiplications first, and then I can add the products together. The multiplications are $4\cdot4 = 16$ and $3\cdot2=6$, so

The same rule is true when division or subtraction is involved: 

This method might seem a bit random, while the ``left to right" method seems more logical. However, the ``multiplication first" method is what people mean when they write things like $4\cdot4-6/2$, and that is how you should calculate it.

I have five pencil cases. Each case contains $2$ red pens, $1$ green pen and $1$ yellow pen. How many pens are there in total? 

To answer this question, it seems like the best way is to first find out how many pens there are in one case. Then we multiply that value by $5$, since there are five cases. So the number of pens is $2+1+1\cdot5$. But that isn't what we mean! Because of the ``multiplication first", this means two, plus one, plus the product of one and five:

Instead we want to sum $2$, $1$ and $1$ before we multiply by $5$. This is where parentheses come in. If we put parentheses around something, that means that we want the things inside the parentheses to be calculated first, even if this breaks the ``multiplication and division first" rule. We can therefore express the number of pens as$$\boldsymbol{(}2+1+1\boldsymbol{)}\cdot5.$$

Earlier in the lesson I made this illustration of $(2+1)\cdot3$:

But we could also view this as $2\cdot3 + 1\cdot3$:

This means that $(2+1)\cdot3 = 2\cdot3 + 1\cdot3$, since they both represent the same circles. And this rule is true in general! As soon as we multiply a whole parentheses, we can instead multiply each term inside the parentheses. This is called distribution.

There is a gap in the rule I have taught so far. See if you can find it yourself by calculating $12/3\cdot2$ in different ways.

Can you see it? $(12/3)\cdot2 = 4\cdot2 = 8$, but $12/(3\cdot2)= 12/6 =2$. When there is only multiplication the order doesn't matter, but when there's also division we can get different answers. The rule I have taught so far doesn't say which of these is correct.

If you type $12/3\cdot2$ into a calculator it will calculate $(12/3)\cdot2$. However, people can't quite agree whether this is right. Therefore, to avoid confusion, the safest thing is to always use parentheses. 

A different notation that avoids parentheses is $12/(3\cdot2) = \frac{12}{3\cdot2}$ and $(12/3)\cdot2 = \frac{12}{3}\cdot2$. Everything below the line divides everything above the line, even without parentheses. This also works for addition: $12/(3+2) = \frac{12}{3+2}$. 

This still isn't the whole story. For example, $4^2$ is a shorthand for $4\cdot4$, but does $-4^2$ equal $(-4)^2$ or $-(4^2)$? And later on you will learn about something called cosine, but does $\cos x+1$ equal $(\cos x) +1$ or $\cos(x+1)$? Usually there's a clear answer, but sometimes not. Then parentheses can be important, to show what you really mean. This isn't anything you need to worry about though. For now, you know everything you need to know about the order of operations.