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What is the order of operations?

You work through an expression in four passes: parentheses first, then exponents, then multiplication and division together from left to right, then addition and subtraction together from left to right. Multiplication and division sit at the same level, so in 24 Γ· 4 Γ— 2 you read left to right and get 12.

The order of operations in plain words

When an expression carries more than one operation, the order of operations decides which comes first. It works in four passes, and you finish each pass before starting the next. Without a fixed order, 8 βˆ’ 2 Γ— 3 has two possible answers; with the order it has one: 2.

  • Parentheses first, from the inside out: 2 Γ— (3 + 4) is 2 Γ— 7, or 14.
  • Exponents next, because a power is settled before the multiplication around it: 2 Γ— 32 is 2 Γ— 9, or 18.
  • Multiplication and division next, together, from left to right.
  • Addition and subtraction last, together, from left to right.

Sharing a level is what catches people. Multiplication does not always come before division, and addition does not always come before subtraction. In 24 Γ· 4 Γ— 2 both signs sit on one level, so you read from the left: 24 Γ· 4 is 6 and 6 Γ— 2 is 12.

Multiply first, as a mnemonic suggests, and you reach a wrong answer: 4 Γ— 2 is 8, and 24 Γ· 8 is 3. Nothing in the notation asked you to put the division off until after the multiplication.

A mnemonic such as PEMDAS misleads on exactly that point. Its letters put M before D and A before S, which reads like a ranking the expression never had. The same trap opens on the other pair: 10 βˆ’ 4 + 3 is 9, because 10 βˆ’ 4 is 6 and 6 + 3 is 9, while subtracting the sum gives 3.

Integers include the negative ones, and a minus sign in front of a number is not a subtraction. In βˆ’7 + 2 Γ— (βˆ’9) both minus signs belong to the numbers, so nothing is subtracted and multiplication still comes first.

ETS lists solving problems with integers using the order of operations among the topics for the Mathematics subtest (5759), where an on-screen four-function calculator is provided.

Why a paraeducator uses this in class

Two errors keep coming back in a classroom, and this skill answers both. One student works strictly from left to right, so 8 βˆ’ 2 Γ— 3 comes out as 18. Another hears a mnemonic and multiplies before dividing, so 24 Γ· 4 Γ— 2 comes out as 3. Both are trying hard, and a red mark teaches neither of them anything.

Knowing the order is also how you find the step that went wrong. When a student hands you a final answer, ask for the work. If that answer matches a different order, the error is in the order, and you can point at the line where the paths part.

A worked example

Take 4 βˆ’ 12 Γ— (3 βˆ’ 5) and read it in four passes.

  • Parentheses: 3 βˆ’ 5 is βˆ’2, so the line becomes 4 βˆ’ 12 Γ— (βˆ’2).
  • Exponents: there are none, so the line does not change.
  • Multiplication and division: 12 Γ— (βˆ’2) is βˆ’24, so the line becomes 4 βˆ’ (βˆ’24).
  • Addition and subtraction: subtracting a negative adds, so 4 βˆ’ (βˆ’24) is 4 + 24, which is 28.

The answer is 28. The last pass is worth pausing on, because subtracting a negative is the same as adding it.

The same order with an exponent: 2 Γ— 32 βˆ’ 10 Γ· 5 becomes 2 Γ— 9 βˆ’ 10 Γ· 5 after the exponent, then 18 βˆ’ 2 when multiplication and division run together from the left, and the last pass gives 16. Both 2 Γ— 9 and 10 Γ· 5 are finished in the multiplication and division pass, before the subtraction.

Practice questions

Question 1: the mnemonic trap

Ana says multiplication always comes before division, so she works out 24 Γ· 4 Γ— 2 as 24 Γ· 8. What is the correct answer, and where did she go wrong?

Show the answer

24 Γ· 4 is 6, and 6 Γ— 2 is 12, so the answer is 12.

The two signs share one level, so the expression is read from the left. Ana reached 3 by multiplying first.

Question 2: a negative integer

Work out βˆ’7 + 2 Γ— (βˆ’9).

Show the answer

The answer is βˆ’25.

The multiplication comes first: 2 Γ— (βˆ’9) is βˆ’18, and then βˆ’7 + (βˆ’18) is βˆ’25. Both minus signs belong to the numbers, so nothing is subtracted.

Question 3: parentheses first

Work out (12 βˆ’ 7)2 + 3 Γ— 2.

Show the answer

The answer is 31.

Parentheses come first, so 12 βˆ’ 7 is 5 and the expression becomes 52 + 3 Γ— 2. The exponent gives 25, multiplication gives 6, and 25 + 6 is 31.

Continue learning

Try it on a full practice test

Once the four passes feel routine, try the free ParaPathways practice test.

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