So why does the order of operations matter? We can use the basic operations models to illustrate just why it makes a difference what we do first, second and so on when we are computing with different operations. Consider the model 3 + 4 x 2.
Here's two ways of looking at it:
We get two different answers. Both are logical and follow the models for basic operations, but you can see why this might cause problems. Mathematicians have decided to agree that we should multiply before adding or subtracting to avoid this confusion. It's just something that we agree to do.
If we wish to add or subtract before multiplying, then we can indicate that with parentheses. (3 x 4) + 3 means multiply first - although we do not necessarily need the parentheses in this case - and 3 x (4 + 2) means to add 4 + 2 first, then multiply by 3. Again, we have made an agreement that if something is in parentheses, it should be done first.
There are other rules for the order of operations, as well. Here's an example with division, 15 ÷ 3 + 2:
Now, let's do the addition first...
Again, both answers are logical and faithful to the models. Here, too, we have just decided to agree that we will do the division first, before adding or subtracting, unless otherwise indicated with parentheses.
We've just seen a couple of examples that illustrate the importance of these agreements, known as the order of operations. To summarize:
These simple rules help us communicate our mathematical thinking around the world, without confusion. They also play an important part in Algebra, as do the models we've seen so far in the first two lessons on this web site. In future lessons, we'll talk about exponents.