Grouping Symbols Handout
PA_M1_S4_T1
The order in which we evaluate mathematical expressions can be ambiguous
and can give me different answers depending on what I do first.
Let's look at this example. I'm going to start with the
left most term and just proceed from left to right
performing the operations as they show up. 3 plus 5
gives me 8, times 9 gives me 72.
It's a valid answer, it looks great, but what
happens if instead of starting with the addition
I start with the multiplication? 3 plus 5 times
9 is 45, and then I add 3 to 45 and I get 48.
Both of these answers look great. Which one is going to be right? And the
right answer is going to depend on what everyone agrees to be the best way
to approach these problems.
It's much easier, when we see things like that, if
grouping symbols show up for us. So let's look at
this same problem and let's look at it with grouping
symbols inserted around the (3 + 5). When I have
grouping symbols it clearly says "here's where you
start." So in this case with grouping symbols in
place, I am first going to compute the sum of 3 and
5 and then multiply by 9 to get 72.
Now sometimes grouping symbols aren't quite
that simple. Sometimes I might have
something that looks like this: [(3 + 5) +
1]*2. These are called nested grouping
symbols and when I see this I'm going to go
to the innermost set of grouping symbols
first and start there. 3 plus 5 is 8, plus
1. And notice that when I evaluate what's
inside the grouping symbols, those grouping symbols go away, and I'm only
left with the grouping symbol on the outside.
Now I'm going to evaluate what's inside the square
bracket grouping symbols. That becomes 9, times 2.
And notice once again, once I've evaluated what's
inside, the grouping symbol drops away, and I get 18.
So this is how we want to approach grouping symbols. When you're setting
up and working your own equations, it's a good idea to use grouping
symbols so that you clearly communicate to other people what it is you
expect to happen first.
If the expression is already given to you with grouping symbols, then you
want to proceed by evaluating what's in the grouping symbols first before
you do anything else.