# Grouping in Addition and Subtraction Problems

## Grouping with parentheses

Sometimes we use parentheses to indicate the order in which arithmetic is to be performed. For example, when you see an arithmetic problem like \$4-(2+3)\$, you should first perform the addition in the parentheses (reducing the problem to \$4-5\$), and then do the remaining subtraction (giving an answer of \$-1\$).

Do each of these arithmetic problems which use parentheses. The problem is pictured on the grid.

In each row of the table below, there are two addition problems with parentheses. Compute each of these sums, performing the addition in the parentheses first. Use the button to switch between the two problems on the grid.

Notice that, when you’re adding three numbers, it doesn’t matter where you put the parentheses. This fact can be written using variables as:

The associative law of addition: for any three numbers \$a\$, \$b\$, and \$c\$, it is always true that
\$\$ (a+b)+c = a+(b+c) \$\$

This is why you can leave out parentheses when writing addition problems, writing things like \$1+2+3\$ instead of \$(1+2)+3\$.

## Parentheses and subtraction

In each row of the table below, there are two subtraction problems with parentheses. Do each of these subtraction problems. Use the button to switch between the problems. (When you look at the picture, remember that subtracting green squares is the same as adding pink squares.)

 Look at any row in the table above. Are the two answers in that row the same? (Type yes or no.)
 Is there an associative law for subtraction? Do you get the same answer, no matter where you put the parentheses? (Type yes or no.)

What about if you have an arithmetic problem that uses subtraction, and doesn’t have parentheses? It should be done from left to right. For example, \$8-5-2\$ always means \$(8-5)-2\$, not \$8-(5-2)\$.

Do these arithmetic problems. Remember to work from left to right.

## Simplifying expressions using the associativity of addition

In each row of the table below, enter the values of the expressions \$(2x+3)+2\$, \$2x+(3+2)\$, and \$2x+5\$ for the given value of \$x\$. These expressions are all illustrated by the grid to the left.

\$x\$\$\cl"tight"{\table {(2x+}, {3)+2}; 2x+, (3+2); 2x+, 5}\$

You can use the associative law of addition to regroup and simplify expressions. That is, you can turn an expression like \$(2x+3)+2\$ into an expression like \$2x+5\$, which is always equal to the original expression no matter what \$x\$ is.

Each row of the table below has some expression involving a variable, addition, and parentheses. Use the associative law of addition to regroup and simplify the expression, as shown in the example in the first row.

## Simplifying expressions with subtraction

There is no associative law for subtraction. So, if you want to use associativity to regroup and simplify an expression that uses subtraction, you need to first convert the subtraction into addition. For example, you can rewrite \$2x-4\$ as \$2x+(-4)\$.

Rewrite each of these expressions involving subtraction so that they involve addition instead.

 The grid to the left shows the expression \$3x+0\$. What is a simpler way of writing this expression?

Each row of the table below has some expression involving a variable, subtraction, and parentheses. By converting the subtraction into addition and using the associative law of addition, regroup and simplify the original expression.