Order of operations

Here you will learn about order of operations, including what it means and how to calculate and solve order of operations problems using PEMDAS.

Students will first learn about order of operations as a part of operations and algebraic thinking in elementary school.

What is the order of operations?

Order of operations refers to the rule that explains the sequence of steps necessary for correctly evaluating a mathematical expression or math problem.

You will use the acronym PEMDAS to help recall the correct order, or priority, in which you complete mathematical operations.

Mathematical operations such as multiplication and addition have to be completed in a specific order. This sequence of steps helps us to evaluate any mathematical expression, both with numerical values and algebraic expressions.

To evaluate an expression using PEMDAS, you need to understand what PEMDAS represents and be able to apply the PEMDAS rule to any calculation.

PEMDAS stands for:

Parentheses, Exponents, Multiplication and Division, Addition and Subtraction

Parentheses have a higher priority than exponents, so we calculate what is inside a pair of parentheses first. Exponents have a higher priority than division and multiplication, so any exponent that can be evaluated is calculated next, and so on.

The order can be remembered using the mnemonic device for PEMDAS, ‘Please Excuse My Dear Aunt Sally.’

Order of operations

Priority of OperationPEMDASMathematical Symbol
1 Parentheses
Order of operations table image 1
2 Exponents
Order of operations table image 2
3 Division


Multiplication

Order of operations table image 3

Order of operations table image 4
4
Addition


Subtraction


Order of operations table image 5

\bf{−}

It is important to note that multiplication and division are given equal priority, and addition and subtraction are given equal priority.

When completing calculations that involve multiplication and division or addition and subtraction, you work from left to right.

For example,

Consider the following expression 12-7+6.

12-7=5 and then 5+6=11

For example,

Consider the following expression 10 \div 5 \times 2.

10 \div 5=2 and then calculate 2 \times 2=4.

Visually you could represent PEMDAS as:

Order of Operations image 2

What is the order of operations?

What is the order of operations?

Common Core State Standards

How does this relate to 5th grade math and 6th grade math?

  • Grade 5: Operations and Algebraic Thinking (5.OA.1)
    Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

  • Grade 5: Operations and Algebraic Thinking (5.OA.2)
    Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

  • Grade 6: Expressions and Equations (6.EE.1)
    Write and evaluate numerical expressions involving whole-number exponents.

How to evaluate using order of operations

In order to evaluate expressions using order of operations, you would use PEMDAS:

  1. Solve any calculations within parentheses.
  2. Solve for any exponents.
  3. Solve any division and multiplication calculations.
  4. Solve any addition and subtraction calculations.

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Use this quiz to check your grade 3 to 6 students’ understanding of properties of equality. 10+ questions with answers covering a range of 3rd, 5th and 6th grade properties of equality topics to identify areas of strength and support!

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Order of operations examples

Example 1: PEMDAS with addition and multiplication

Evaluate 3+6\times{7}.

  1. Solve any calculations within parentheses.

There are no parentheses.

2Solve for any exponents.

There are no exponents.

3Solve any division and multiplication calculations.

The multiplication that we need to calculate is 6\times{7}=42.

Replacing 6\times{7} with 42 gives us the calculation 3+42.

4Solve any addition and subtraction calculations.

3+42=45

So 3+6\times{7}=45

Example 2: PEMDAS with division and subtraction

Evaluate 12-8\div{2}.

Solve any calculations within parentheses.

Solve for any exponents.

Solve any division and multiplication calculations.

Solve any addition and subtraction calculations.

Example 3: PEMDAS with parentheses and multiplication

Evaluate 3(2+4\times{6}-3).

Solve any calculations within parentheses.

Solve for any exponents.

Solve any division and multiplication calculations.

Solve any addition and subtraction calculations.

Example 4: PEMDAS with an exponent and multiplication

Evaluate 4\times{3}^{2}.

Solve any calculations within parentheses.

Solve for any exponents.

Solve any division and multiplication calculations.

Solve any addition and subtraction calculations.

Example 5: PEMDAS with brackets, an exponent and division

Evaluate 3+(10\div{4}\times{20})^{2}.

Solve any calculations within parentheses.

Solve for any exponents.

Solve any division and multiplication calculations.

Solve any addition and subtraction calculations.

Example 6: PEMDAS with everything

Evaluate 4^{2}+2(14-8)\div{3}.

Solve any calculations within parentheses.

Solve for any exponents.

Solve any division and multiplication calculations.

Solve any addition and subtraction calculations.

Teaching tips for order of operations

  • While PEMDAS is a great way for students to remember the order of operations, it’s important that the students understand how to solve for each of the operations required, not simply giving them the acronym to memorize.

  • It is important to find a way for students to practice outside of order of operations worksheets. The more practice a student gets, the better. For example you can physically separate word problems from the answer choices in the classroom. Students would have to walk around to discuss and solve the problem and then find the answer that matches.

  • Students could be given an expression without the symbols. They would have to then decide which operations/symbols would make the expression true.

Easy mistakes to make

  • Multiplication/division and addition/subtraction in the incorrect order
    When you have a chain of multiplication and division calculations or a chain of addition and subtraction calculations, you work from left to right.
    For example, If you put parentheses into the calculation 12\div{4}\times{3} with the purpose of keeping the calculation exactly the same, you must place the parentheses around 12\div{4}.
    This is because 12\div{4}\times{3}=9, and (12\div{4})\times{3}=3\times{3}=9.
    If you placed parentheses around 4\times{3} you would have 12\div{(4\times{3})}=12\div{12}=1 which is a different answer that you would get without the parenthesis.

  • Exponents as multiplying a number by 2
    When introducing exponents, some students will interpret 4^2 as 4 \times 2 instead of 4 \times 4. The use of a number line to represent this expression or using expanded notation when introducing the concept can help students internalize the learning.

  • Multiplying before exponents
    Take 2\times{3}^{4}. Because you would perform exponents before you perform multiplication within PEMDAS, you need to calculate the exponent term first, and then multiply the solution by 2. Inserting a pair of parentheses with the purpose of keeping the calculation exactly the same, 2\times{3}^{4}=2\times(3^{4})=162. A common misconception is that the multiplication is carried out first, and then the value is raised to the power; here this would look like (2\times{3})^{4} and is equal to 6^{4}=1296 which is the wrong answer.

Practice order of operations questions

1. Calculate 6+5 \times 8.

88
GCSE Quiz False

46
GCSE Quiz True

640
GCSE Quiz False

118
GCSE Quiz False

To solve, follow the order of operations by using PEMDAS.

 

Order of Operations practice question 1

2. Calculate 16-15 \div 3.

0.\dot{3}
GCSE Quiz False

1.3
GCSE Quiz False

9
GCSE Quiz False

11
GCSE Quiz True

To solve, follow the order of operations by using PEMDAS.

 

Order of Operations practice question 2

3. Calculate (8-2 \times 5+4) \div 2.

17
GCSE Quiz False

1
GCSE Quiz True

2
GCSE Quiz False

11
GCSE Quiz False

To solve, follow the order of operations by using PEMDAS.

 

Order of Operations practice question 3

4. Calculate 2 \times 5^2.

20
GCSE Quiz False

100
GCSE Quiz False

50
GCSE Quiz True

49
GCSE Quiz False

To solve, follow the order of operations by using PEMDAS.

 

Order of Operations practice question 4

5. Calculate 258-(10 \div 2 \times 3)^2.

33
GCSE Quiz True

255
GCSE Quiz False

228
GCSE Quiz False

43
GCSE Quiz False

To solve, follow the order of operations by using PEMDAS.

 

Order of Operations practice question 5

6. Calculate 3(7-5)^3 \div 6+4.

2
GCSE Quiz False

7
GCSE Quiz False

8
GCSE Quiz True

12
GCSE Quiz False

To solve, follow the order of operations by using PEMDAS.

 

Order of Operations practice question 6

Order of operations FAQs

Do you perform multiplication or division first in order of operations?

In the order of operations, multiplication and division are seen as equal and would be performed as they are stated within the expression, starting from left to right. The same is true with addition and subtraction.

Is PEMDAS or BODMAS correct?

Both PEMDAS and BODMAS are correct. The acronym BODMAS is the UK version of the same rules. It can also be referred to as BIDMAS or BEDMAS. The acronym in the UK would read as: Brackets, Indices, Division and Multiplication, Addition and Subtraction.

What is an exponent?

An exponent refers to the number of times a number is multiplied by itself. For example 7^2 = 7 \times 7 or 4^3 = 4 \times 4 \times 4.

Why do we use parentheses in math?

Parentheses are used within mathematical expressions to note a modification is the normal order of operations. In the expression, (3+4) \times 3, because the addition is within parentheses, it is solved before the multiplication.

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