5th Grade · 5.OA.A.1

Order of operations in 5th grade

The order of operations is the agreed order for the steps in a math problem, such as $6 + 4 \times 5$ . In 5th grade, your child uses the order of operations with parentheses, brackets and braces, including expressions where one set of symbols sits inside another.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

What it means, in plain words

An operation is adding, subtracting, multiplying or dividing. When a problem has more than one, the order changes the answer. $6 + 4 \times 5$ could be 50 if you add first, or 26 if you multiply first. Everyone agrees on 26, so the order of operations says to multiply before you add.

The full order is: work inside grouping symbols first, then multiply and divide from left to right, then add and subtract from left to right. Grouping symbols come in three shapes, parentheses ( ), brackets [ ] and braces { }. Order of operations with parentheses comes first in 5th grade. Then children see brackets and braces around the parentheses. When symbols sit inside each other, start with the innermost set and work outward.

Many parents learned the order as PEMDAS. The memory trick has one trap. Multiplication and division are equal partners, done in order from left to right, and so are addition and subtraction. The E, for exponents, waits until 6th grade. In 5th grade, exponents only show up as powers of 10, like $10^3$ .

Your child should be able to:

Evaluate an expression with parentheses, brackets and braces, starting from the inside
Multiply and divide from left to right before adding and subtracting
Place parentheses to change the value of an expression
Explain why $6 + 4 \times 5$ is 26 and not 50

Example

Evaluate $2 \times (9 + 6) - 10$ .

Parentheses first: $9 + 6 = 15$ .
Then multiply: $2 \times 15 = 30$ .
Then subtract: $30 - 10 = 20$ .
The value of the expression is 20.

Example

Evaluate $40 - \{3 \times [10 - (2 + 4)]\}$ .

Start with the innermost parentheses: $2 + 4 = 6$ .
Then the brackets: $10 - 6 = 4$ .
Then the braces: $3 \times 4 = 12$ .
Last, subtract: $40 - 12 = 28$ .
The value of the expression is 28.

Where children usually get stuck

Your child works out $12 + 8 \div 4$ as $20 \div 4 = 5$ instead of 14.
Your child reads the problem left to right like a sentence and does each step in the order it appears.
What helps. Before any arithmetic, have your child circle every multiplication and division sign and do those steps first. Here $8 \div 4 = 2$ , and $12 + 2 = 14$ .
Your child works out $24 \div 4 \times 2$ as $24 \div 8 = 3$ instead of 12.
PEMDAS lists M before D, so the child multiplies first. Multiplication and division are one step, done from left to right.
What helps. Write the trick on paper as P, then MD on one line, then AS on the next. Then work left to right: $24 \div 4 = 6$ , and $6 \times 2 = 12$ .
Your child works out $5 \times [14 - (3 + 7)]$ as $70 - 10 = 60$ .
The 5 sits next to the bracket, so the child multiplies it by the first number inside and forgets that the whole bracket is one amount. The right answer is $5 \times 4 = 20$ .
What helps. Have your child work out the innermost part, then rewrite the whole line with the new number before going on. One step per line keeps the brackets together.

Try four practice questions

Real 5th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Drag both parentheses into the blanks to make the equation true.
Tap or click a tile to drop it into the highlighted blank, or tap an empty blank first to choose a different one. Tap a filled blank to send its tile back.
3
+
4
×
2
=
14
(
)
Tap an item to place it in the next empty slot, or drag it to a specific slot. With a keyboard: select an item with Enter or Space, then select a slot to place it. Press Delete or Backspace on a slot to remove an item.
Show the answer and how to solve it
Put ( before 3 and ) after 4. Then the expression becomes $(3 + 4) \times 2$ , which equals 14.
1. 1First evaluate the expression without any parentheses. Multiplication happens before addition, so $4 \times 2$ is $8$ , giving $11$ . That is not the target $14$ .
2. 2Try grouping $3 + 4$ first: the result is $14$ , which matches the target. Grouping $4 \times 2$ does not change anything because multiplication was already done first, giving $11$ again.
3. 3So the opening parenthesis goes before 3 and the closing parenthesis goes after 4.
Common mistake. A common mistake is putting parentheses around $4 \times 2$ . That does not change the order of operations, so the expression still equals $11$ instead of $14$ .
The idea underneath. Parentheses only change the value when they make you do a different part of the expression first.
2On level
Maya is finding the value of $4 + 3 \times (9 - 4)$ . Her work is shown below.
Start
$\displaystyle 4 + 3 \times (9 - 4)$
After parentheses
$\displaystyle = 4 + 3 \times 5$
Student's last step
$\displaystyle = 4 + 15 = 19$
Select the statement that best describes the steps Maya wrote.
A
Maya found the value inside the parentheses incorrectly.
B
Maya's work is correct.
C
Maya found the value inside the parentheses correctly, but then used the wrong order of operations afterward.
D
Maya did not evaluate the expression inside the parentheses first.
Show the answer and how to solve it
Maya's work is correct. The expression $4 + 3 \times (9 - 4)$ equals $19$ .
1. 1Look at Maya's first change. She changed $9 - 4 = 5$ , so she evaluated the parentheses correctly.
2. 2After the parentheses, the expression should be $4 + 3 \times 5$ .
3. 3Multiply before you add: $3 \times 5 = 15$ .
4. 4Then add $4 + 15$ to get the correct value of $19$ .
5. 5Maya's last line shows $4 + 15 = 19$ , which matches the correct path. Every step she wrote is correct.
Common mistake. Checking only the final value can hide a mistake. Maya's written steps show parentheses first, then multiplication, then addition.
The idea underneath. Parentheses tell you what to do first, but they do not cancel the rest of the order of operations.

3On level

Evaluate the expression below by working through each step. Apply the order of operations: parentheses first, then multiplication or division, then addition or subtraction.

\displaystyle 8 + 3 \times (7 - 4)

Compute each step and type the result in the box.

Original	$\displaystyle 8 + 3 \times (7 - 4)$
Step 1: parentheses	$\displaystyle = 8 + 3 \times$
Step 2: multiplication	$\displaystyle = 8 +$
Step 3: addition	$\displaystyle =$

Show the answer and how to solve it

Compute the parentheses to get $3$ , then multiply $3 \times 3 = 9$ , then add $8 + 9 = 17$ . The value of the expression is $17$ .

1Apply the order of operations: parentheses first, then multiplication, then addition.
2Step 1 is the parentheses: $7 - 4 = 3$ . Type $3$ into the first box.
3Step 2 is the multiplication. Substitute $3$ for the parentheses and compute $3 \times 3 = 9$ . Type $9$ into the second box.
4Step 3 is the addition. Add $8 + 9 = 17$ . Type $17$ into the third box.

Common mistake. A common mistake is to add $8$ and $3$ before multiplying. After you solve the parentheses, $3 \times 3$ must be done before you add $8$ .

The idea underneath. Parentheses are evaluated first. After the parentheses are gone, multiplication and division still come before addition and subtraction.

4Stretch

A student evaluated the numerical expression below. Their work is shown step by step, but exactly one step contains a mistake.

Start	$\displaystyle 24 - [12 \div (8 - 2) + 7]$
Step 1	$\displaystyle = 24 - [12 \div 6 + 7]$
Step 2	$\displaystyle = 24 - [2 + 7]$
Step 3	$\displaystyle = (24 - 2) + 7$
Step 4	$\displaystyle = 29$

Select the statement that correctly names the step with the mistake and explains what the student did wrong at that step.

Step 1: the student should have subtracted 24 minus 12 before simplifying anything inside the brackets.

Step 2: the student divided 12 by 6 to get 2 instead of first adding 7 inside the brackets.

Step 3: the student dropped the brackets and subtracted 2 from 24 before adding 7, but the addition inside the brackets had to be finished first.

Step 4: the student added 22 and 7 and got the wrong sum.

Show the answer and how to solve it

The mistake is in Step 3: the student dropped the brackets and subtracted $2$ from $24$ before adding $7$ , giving $29$ instead of the correct value $24 - [12 \div (8 - 2) + 7] = 24 - [2 + 7] = 24 - 9 = 15$ .

1Start with the original expression $24 - [12 \div (8 - 2) + 7]$ . The innermost group is $(8 - 2)$ , so that group must be simplified first.
2Replace $(8 - 2)$ with $6$ . Then divide inside the brackets: $12 \div 6 = 2$ . The brackets still contain $2 + 7$ , which has not been simplified yet.
3Finish the work inside the brackets: $2 + 7 = 9$ , so the expression becomes $24 - 9$ . This is the step where the student went wrong: instead of adding $7$ inside the brackets first, the student dropped the brackets and subtracted $2$ from $24$ to get $(24 - 2) + 7$ , which changes the order of operations.
4Now do the outside subtraction: $24 - 9 = 15$ . The correct value of the original expression is $15$ , not $29$ .

Common mistake. A common mistake is dropping the brackets too early and doing the outside subtraction before the bracketed expression is fully simplified. In this problem, the student went from $24 - [2 + 7]$ to $(24 - 2) + 7$ , which evaluates to $29$ instead of the correct value $15$ .

The idea underneath. Brackets group an operation that must be finished before any outside operation is done. When you replace the innermost group with a number, the brackets stay until the bracketed expression itself is fully simplified.

Start	$\displaystyle 4 + 3 \times (9 - 4)$
After parentheses	$\displaystyle = 4 + 3 \times 5$
Student's last step	$\displaystyle = 4 + 15 = 19$