Parentheses, Brackets, Braces and Spotting Mistakes | Grade 5 Math | LevelKite

Three friends go to a fair. Each friend pays 8 dollars for a ticket and 2 dollars for a snack. How much do they pay in all?

Each friend pays $8 + 2$ dollars, and there are 3 friends. So you need to add first and then multiply by 3.

Suppose you write $3 \times 8 + 2$ . This is an expression: numbers and operation signs, such as $+$ and $\times$ , written together with no equals sign. To evaluate an expression means to find its value. You already know that multiplying comes before adding. So this expression multiplies $3 \times 8$ first, and the snack is added only once.

To keep $8 + 2$ together, put curved marks called parentheses around it and write $3 \times (8 + 2)$ . Parentheses are grouping symbols. Grouping symbols come in pairs. They hold part of an expression together as one amount, and they tell you to find the value of that part first.

The table shows what each expression counts.

Only the second expression counts a snack for every friend, so the friends pay 30 dollars.

Expression	What it counts	Value
$3 \times 8 + 2$	3 tickets and only 1 snack	$24 + 2 = 26$ dollars
$3 \times (8 + 2)$	3 friends, each with 1 ticket and 1 snack	$3 \times 10 = 30$ dollars

Worked example

Parentheses first

Evaluate $25 - (4 + 2) \times 3$ .

Look for grouping symbols. The parentheses hold 4 + 2 together as one amount, so find that value first: 4 plus 2 is 6. Write 6 in place of the whole group and copy the rest unchanged.
$25 - (4 + 2) \times 3 = 25 - 6 \times 3$
No grouping symbols are left, so use the order you already know. Multiplying comes before subtracting, so multiply 6 by 3 to get 18.
$25 - 6 \times 3 = 25 - 18$
Only a subtraction is left. Take 18 away from 25.
$25 - 18 = 7$

Answer

The value of $25 - (4 + 2) \times 3$ is $7$ .

Sometimes one group sits inside another group. If every group used parentheses, it would be hard to see where each group starts and ends. So there are two more kinds of grouping symbols. Brackets are square, as in $[10 - 6]$ . Braces are curly, as in $\{8 + 12\}$ . They do the same job as parentheses.

Most often, parentheses go on the inside, brackets go around them, and braces go around the brackets.

A group with no other group inside it is called the innermost group. You start there. Why? The inner group is one of the amounts the outer group works with. You cannot finish the outer group until the inner group is a single number.

The labels below show the order for one expression.

60 - \underbrace{\{\,8 + 3 \times \underbrace{[\,10 - \underbrace{(5 + 1)}_{\text{first}}\,]}_{\text{second}}\,\}}_{\text{third}}

Key concept

Work from the inside out

Grouping symbols hold part of an expression together as one amount. You need that amount before you can use it. So:

Find the value of the innermost group first.
Move outward one group at a time. Most often that means parentheses, then brackets, then braces.
Inside each group, use the order you already know. Multiply and divide from left to right. Then add and subtract from left to right. Use the same order after all the groups are gone.

Each time you finish a group, write its value in place of the whole group and copy everything else unchanged.

Worked example

Parentheses, brackets and braces

Evaluate $60 - \{8 + 3 \times [10 - (5 + 1)]\}$ , the expression labeled above.

Start with the innermost group. The parentheses have no other group inside them. Add 5 and 1 to get 6, and write 6 in place of the parentheses.
$60 - \{8 + 3 \times [10 - (5 + 1)]\} = 60 - \{8 + 3 \times [10 - 6]\}$
Move out to the brackets. With the parentheses gone, the brackets are now the innermost group. Take 6 away from 10 to get 4.
$60 - \{8 + 3 \times [10 - 6]\} = 60 - \{8 + 3 \times 4\}$
Move out to the braces. Inside them are an addition and a multiplication. Multiplying comes before adding, so multiply 3 by 4 to get 12.
$60 - \{8 + 3 \times 4\} = 60 - \{8 + 12\}$
Only an addition is left inside the braces. Add 8 and 12 to get 20, and the braces are done.
$60 - \{8 + 12\} = 60 - 20$
No groups are left, so subtract.
$60 - 20 = 40$

Answer

The value of $60 - \{8 + 3 \times [10 - (5 + 1)]\}$ is $40$ .

An expression has only one value. So if two people evaluate the same expression and get different values, at least one of them made a mistake. Here is how to find a mistake in someone's work:

Read the work one line at a time, starting at the top.
For each line, ask what should happen next. The innermost group comes first. Inside a group, or when all the groups are gone, multiplying and dividing come before adding and subtracting.
Stop at the first line that does something else. That line holds the mistake.
Correct that line, then finish the work from there.

Look at $5 + 2 \times [10 - (3 + 1)]$ . Worked from the innermost group outward, it goes $3 + 1 = 4$ , then $10 - 4 = 6$ , then $2 \times 6 = 12$ , then $5 + 12 = 17$ . Its value is $17$ . The table shows three common mistakes on this same expression.

Three common mistakes on the same expression

Mistake	The mistaken work	Wrong value	How to correct it
Working strictly from left to right	$5 + 2 = 7$ , then $7 \times 10 = 70$ , then $70 - 3 = 67$ , then $67 + 1 = 68$	$68$	Start with the innermost group, $(3 + 1)$ , not with the first operation on the left.
Adding before multiplying	$3 + 1 = 4$ , then $10 - 4 = 6$ , then $5 + 2 = 7$ , then $7 \times 6 = 42$	$42$	After the groups, multiply $2 \times 6$ first. Then add 5.
Leaving an inner group for later	$10 - 3 = 7$ , then $7 + 1 = 8$ , then $2 \times 8 = 16$ , then $5 + 16 = 21$	$21$	Find the inner group $(3 + 1)$ before you subtract inside the brackets.

Worked example

Find and correct a mistake

A classmate evaluates $\{30 - [12 - (3 + 5)]\} \times 2$ . The classmate writes $\{30 - [12 - 8]\} \times 2$ , then $\{18 - 8\} \times 2$ , then $10 \times 2$ , then $20$ . Find the mistake and correct it.

Check the first line. The parentheses are the innermost group, and 3 plus 5 is 8. The first line follows the order, so it is correct.
$\{30 - [12 - (3 + 5)]\} \times 2 = \{30 - [12 - 8]\} \times 2$
Check the second line. The brackets are now the innermost group, so 8 should be taken away from 12 next. The classmate took 12 away from 30 instead and left the brackets for later. That is the mistake. The brackets say the amount to take away from 30 is 12 minus 8, which is 4. The classmate took away 12 and then 8 more.
$30 - [12 - 8] \ne 18 - 8$
Correct the second line. The brackets come first: take 8 away from 12 to get 4.
$\{30 - [12 - 8]\} \times 2 = \{30 - 4\} \times 2$
Move out to the braces. Take 4 away from 30 to get 26.
$\{30 - 4\} \times 2 = 26 \times 2$
No groups are left, so multiply 26 by 2.
$26 \times 2 = 52$

Answer

The mistake is in the second line. The classmate subtracted $30 - 12$ before finding the value of the brackets. The correct value is $52$ , not $20$ .

Grouping symbols show which part of an expression acts as one amount. Find the innermost group first and work outward, most often from parentheses to brackets to braces. Inside each group, and when the groups are gone, multiply and divide before you add and subtract. To find a mistake, read the work line by line and stop at the first line that breaks this order. Then correct that line and finish.

Next, you will compare expressions that use the same numbers and operations but have parentheses in different places. You will also place grouping symbols yourself so that an expression has the value you want.