The Order of Operations: One Expression, One Value | Grade 5 Math | LevelKite

An expression is made of numbers and operation signs written in a row, such as $4 + 3 \times 2$ . It has no equals sign. An operation is a way of combining two numbers to make one number. This lesson uses four operations: addition, subtraction, multiplication and division. To evaluate an expression means to find the single number it is worth. That number is called its value.

Here is a puzzle. Maya and Leo both evaluate the expression below. Each of them does two calculations, and every calculation is correct. But they do them in different orders.

4 + 3 \times 2

One expression, evaluated in two orders

Student	First calculation	Second calculation	Value found
Maya adds first	$4 + 3 = 7$	$7 \times 2 = 14$	$14$
Leo multiplies first	$3 \times 2 = 6$	$4 + 6 = 10$	$10$

The same expression gave two values, $14$ and $10$ . Neither student made a calculation slip. The only difference was the order.

That is a real problem. If one expression could be worth two different numbers, you could never be sure what another person meant by it. Nobody could check anyone else's work.

The fix is an agreement. Everyone who uses mathematics follows the same rules about which operation to do first, next and last. These rules are called the order of operations. Because everyone follows them, every expression has exactly one value.

Which order did people agree on? A sticker count shows the idea.

You have $4$ loose stickers and $3$ packs with $2$ stickers in each pack. In symbols, the number of stickers is $4 + 3 \times 2$ .

$3 \times 2$ means $3$ packs of $2$ stickers, which is $6$ stickers.
Now join the loose stickers and the pack stickers: $4 + 6 = 10$ stickers.

Adding $4 + 3$ first would join $4$ stickers to $3$ packs. The $7$ you would get is not a number of stickers. A multiplication such as $3 \times 2$ acts as one amount, so you find it before you add. Division works the same way. The agreed order follows this idea: the value of $4 + 3 \times 2$ is $10$ , as Leo found.

Key concept

The order of operations

When an expression has only numbers and the signs $+$ , $-$ , $\times$ and $\div$ , evaluate it in two passes. A pass is one trip across the expression from left to right.

First pass: multiply and divide, from left to right.
Second pass: add and subtract, from left to right.

Multiplication and division share the first pass. Addition and subtraction share the second pass. Within a pass, the operation farther to the left goes first.

How you write your work matters too. Do one operation on each line.

Write the expression on the first line.
Choose the operation that the order of operations says comes next.
On a new line, write the result in place of the two numbers and the sign you just used. Copy everything else exactly as it was.
Repeat until one number is left. That number is the value.

Your lines together make a record of your work. Each line differs from the line above it in only one place, so the record is easy to check. Read down the lines and ask two questions about each one: Was this the right operation to do next? and Is this one fact correct? If the value is wrong, the record shows you which line went wrong.

Leo's work, recorded with one operation on each line

Line	What you write	Operation done	Fact to check
1	$4 + 3 \times 2$	None yet	The expression is copied correctly
2	$4 + 6$	Multiply $3$ by $2$	$3 \times 2 = 6$
3	$10$	Add $4$ and $6$	$4 + 6 = 10$

Worked example

One division, one subtraction

Evaluate $20 - 12 \div 4$ . Record one operation on each line.

Begin with the first pass: look for multiplication or division. There is one division, 12 divided by 4, and it is 3. It goes before the subtraction, even though the subtraction is farther left, because the first pass comes before the second. Write 3 in its place and copy the 20 and the minus sign.
$20 - 12 \div 4 = 20 - 3$
No multiplication or division is left, so begin the second pass. One subtraction remains: 20 minus 3 is 17.
$20 - 3 = 17$

Answer

The value of $20 - 12 \div 4$ is $17$ .

Worked example

Division and multiplication in the same pass

Evaluate $24 \div 4 \times 3 + 5$ . Record one operation on each line.

First pass: there is a division and a multiplication. They share this pass, so take them from left to right. The division is farther left, so it goes first: 24 divided by 4 is 6. Copy the rest unchanged.
$24 \div 4 \times 3 + 5 = 6 \times 3 + 5$
The first pass is not finished, because a multiplication is left: 6 times 3 is 18. Copy the plus 5.
$6 \times 3 + 5 = 18 + 5$
Second pass: only an addition is left. 18 plus 5 is 23.
$18 + 5 = 23$

Answer

The value of $24 \div 4 \times 3 + 5$ is $23$ .

Worked example

All four operations

Evaluate $250 - 12 \times 15 + 48 \div 6$ . Record one operation on each line, then check the record.

First pass, from left to right: the first multiplication or division you meet is 12 times 15, which is 180. Write 180 in its place and copy everything else.
$250 - 12 \times 15 + 48 \div 6 = 250 - 180 + 48 \div 6$
Still the first pass: 48 divided by 6 is 8. Write 8 in its place.
$250 - 180 + 48 \div 6 = 250 - 180 + 8$
Second pass, from left to right: the subtraction is farther left than the addition, so it goes first. 250 minus 180 is 70.
$250 - 180 + 8 = 70 + 8$
One addition is left: 70 plus 8 is 78.
$70 + 8 = 78$
Check the record. Each line used one fact: 12 times 15 is 180, 48 divided by 6 is 8, 250 minus 180 is 70, and 70 plus 8 is 78. All four facts are correct, and each operation was the right one to do next.

Answer

The value of $250 - 12 \times 15 + 48 \div 6$ is $78$ .

Takeaway: an expression must have exactly one value, so everyone evaluates it in the same agreed order. First multiply and divide from left to right. Then add and subtract from left to right. Write one operation on each line so that every step can be checked.

Next, you will evaluate expressions that have parentheses, brackets and braces. These are symbols that group part of an expression.