1. Order of Operations
By Ashley Bartkowiak
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2. Lesson Overview
Objective: to use the order of operations to simplify numerical expressions
Why do we need to learn about the order of operations?: Like most things in life, we have a set of rules in mathematics that we must follow so that everyone behaves in the same way. Much like a classroom with no rules, without the order of operations, mathematics problem-solving becomes a bit chaotic; one problem could have three different answers. Thus, the order of operations have been established to keep everyone on the same page when simplifying expressions.
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3. Order of Operations Homepage
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4. Vocabulary Terms
Numerical Expression: a mathematical phrase that includes numbers and operational symbols
Example: 3x+2
Operation: a process or action which produces a new value
Examples: multiplication, division, subtraction, addition
Power: the base and the exponent of an expression
Example: 3² (3 is the base and 2 is the
exponent)
*Click on the vocabulary word to take you back to the page with that term.
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5. Simplifying Expressions
To simplify a numerical expression, you replace it with its simplest name.
Example 1: The simplest name for the expression 2+3 is 5 (we add 2 and 3 to get a sum of 5).
Example 2: The simplest name for the expression 3² is 9, (3² is the same thing as 3 × 3 and when we multiply 3 and 3 we get a product of 9).
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*Click on the vocabulary term to take you to the definition.

6. Let’s take a look at the expression 2 + 8 – 6 ÷ 2
Let’s take a look at the expression – 6 ÷ 2. John and Sarah both simplified the expression, but they got two different answers. Although their work is different, they are both convinced that they are correct.
John’s Work: Sarah’s Work:
2 + 8 – 6 ÷ – 6 ÷ 2
10 – 6 ÷ – 3
4 ÷ – 3
Different Answers
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7. To avoid getting different answers when simplifying the same expression (like with John and Sarah), mathematicians have agreed on an order for doing operations. There are essentially 4 steps that we follow to simplify expressions.
Order of Operations
Step 1: Perform any operation(s) inside grouping symbols (grouping symbols include parentheses ( ), brackets [ ], or absolute value signs||). Note: If there are more than one set of grouping symbols, then start with the innermost set.
Step 2: Simplify powers (get rid of the exponents).
Step 3: Multiply and divide in order from left to right.
Step 4: Add and Subtract in order from left to right.
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8. Helpful Tips Parentheses Exponents Multiply Divide Add Subtract
A common technique for remembering the order of operations is to simplify each step to one word and use the abbreviation PEMDAS.
Parentheses
Exponents
Multiply
Divide
Add
Subtract
You can remember PEMDAS by thinking of the phrase “Please Excuse My Dear Aunt Sally.”
Left to Right
Left to Right
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9. Order of Operations
Going back to our original expression – 6 ÷ 2, let’s use the order of operations to find out who (John or Sarah) found the correct answer.
Step 1: There are no parentheses or grouping symbols, so we skip this step and go on to the next.
Step 2: There are no exponents, so we skip this step.
Step 3: We now multiply/divide from left to right. This means that the first thing we need to do is 6 ÷ 2, which gives us an answer of 3. So, when we re-write the expression, we replace 6 ÷ 2 with 3 to get – 3.
Step 4: We now add/subtract from left to right. So, we are going to start by doing the computation 2 + 8, which gives us 10. If we re-write the expression we now have 10 – 3. When we subtract, we get a final answer of 7.
Since we ended up with a final answer of 7, Sarah was correct.
Parentheses
Exponents
Multiply/Divide
Add/Subtract
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10. Example 1: Order of Operations Problem: (17 – 7) ÷ 5 + 1
Solution: ÷ 5 + 1
2 + 1 3
The simplest name for the numerical expression (17 – 7) ÷ is 3.
Parentheses: 17 – 7 = 10
Exponents: No exponents
Multiply/Divide (Left to Right): 10 ÷ 5 = 2
Add/Subtract (Left to Right): = 3
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11. Example 2: Order of Operations Problem: 25 – 8 × 2 + 3²
Solution: – 8 × 2 + 9
25 –
9 + 9 18
The simplest name for the numerical expression 25 – 8 × 2 + 3² is 18.
Parentheses: No parentheses
Exponents: 3² = 3×3 = 9
Multiply/Divide (Left to Right): 8×2 = 16
Add/Subtract (Left to Right): First, 25 – 16 = 9
Then, = 18
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12. Example 3: Order of Operations Problem: 3 + |1 – 2| Solution: 3 + 1 4
The simplest name for the numerical expression 3 + |1 – 2| is 4.
Parentheses: The absolute value signs are treated as grouping symbols, like parentheses. 1 – 2 = -1 and the absolute value of -1 is 1.
Exponents: No exponents
Multiply/Divide (Left to Right): No multiplication or division
Add/Subtract (Left to Right): = 4
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13. Example 4: Order of Operations Problem: 6[13 – 2 (4 + 1)]
Solution: (13 – 2 × 5)
6 (13 – 10)
6 × 3 18
The simplest name for the numerical expression 6[13 – 2 (4 + 1)] is 18.
Parentheses: There are two sets of grouping symbols, so we start with the innermost = 5. Then, go to the second set of parentheses. Treat what is inside (13 – 2 × 5) like a new expression. To simplify, follow the remaining steps (EMDAS).
Exponents: There are no
exponents
Multiply/Divide: 2 × 5 = 10.
Add/Subtract: 13 – 10 = 3. We have taken care of what is in the parentheses and are left with the remaining expression 6 × 3. We can now move on to the step below.
Exponents: No exponents in 6 × 3
Multiply/Divide (Left to Right): 6 × 3 = 18
Add/Subtract (Left to Right): No addition or subtraction
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14. Example 5: Problem: 9 + [4 – (10 – 9)²]³ Order of Operations
Solution: (4 – 1²)³
9 + (4 – 1)³
9 + 3³
9 + 27 36
The simplest name for the numerical expression 9 + [4 – (10 – 9)²]³ is 36.
Parentheses: There are two sets of grouping symbols, so we start with the innermost 10 – 9 = 1. Then, go to the second set of parentheses. Treat what is inside (4 - 1²) like a new expression. To simplify, follow the remaining steps (EMDAS).
Exponents: 1² = 1
Multiply/Divide: No
multiplication or division
Add/Subtract: 4 – 1 = 3
We have taken care of what is in the parentheses and are left with the remaining expression 9 + 3³. We can now move on to the step below.
Exponents: 3³ = 3 × 3 × 3 = 9 × 3 = 27
Multiply/Divide (Left to Right): No multiplication or division
Add/Subtract (Left to Right): = 36
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15. Question 1
Given 3 × 6 – 4² ÷ 2, what is the first step to simplifying the expression? a. b. c.
3 × 6 4²
4² ÷ 2
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16. Try Again! Think of PEMDAS! Parentheses Exponents Multiply/Divide
Add/Subtract
Try Again!
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17. Think Hard! Think of PEMDAS! Parentheses Exponents Multiply/Divide
Add/Subtract
Think Hard!
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18. Way to go!
Following the order of operations (PEMDAS), parentheses come first. However, there are no parentheses in the given expression, so we move on to exponents. That means that we need to start by simplifying 4².
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19. Question 2
Given 2(5 + 9) – 6 , what is the first step to simplifying the expression? a. b. c.
5 + 9
2 × 5
9 – 6
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20. That is correct!
Following the order of operations (PEMDAS), parentheses come first. That means that we need to start by simplifying
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21. Try Again! Think of PEMDAS! Parentheses Exponents Multiply/Divide
Add/Subtract
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22. Sorry! Think of PEMDAS! Parentheses Exponents Multiply/Divide
Add/Subtract
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23. Question 3
Given |7² – 16| ÷ 8 , what will the next step look like when simplifying the expression? a. b. c.
|7²| – 2
49 – 16 ÷ 8
|49 – 16| ÷ 8
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24. Remember, absolute value signs are treated as grouping symbols.
Think Again!
Remember, absolute value signs are treated as grouping symbols.
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25. Did you do all of the operations inside of the grouping symbols before getting rid of them?
Take Another Look!
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26. You Did It!
Following the order of operations (PEMDAS), parentheses come first. That means that we need to start by looking at the expression inside of the parentheses (7² – 16). If we think of this as a new expression, we follow the steps following the parentheses (EMDAS). This means that we start by simplifying the power 7².
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27. Question 4
Given [3(7 + 4) – 2]6, what will you need to do first to simplify the expression? a. b. c.
Simplify what is in the brackets [ ]
Simplify what is in the parentheses ( )
Multiply 11 and 3
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28. Nice Work!
Following the order of operations (PEMDAS), parentheses come first. Since we have two sets of parentheses, we simplify the expression in the innermost set first. That means that we would start with
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29. Not Quite!
When you have more than one set of grouping symbols what do you do first?
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30. That is Incorrect! Think of PEMDAS! Parentheses Exponents
Multiply/Divide
Add/Subtract
That is Incorrect!
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31. Question 5
What is the simplest form of the expression 40 – 2 × 3²? a. b. c. 22
342 28
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32. Remember, 3² is the same as 3 × 3
Keep Trying!
Remember, 3² is the same as 3 × 3
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33. Great Job! Order of Operations Problem: 40 – 2 × 3²
Parentheses: No parentheses
Exponents: 3² = 3 × 3 = 9
Multiply/Divide (Left to Right): 2 × 9 = 18
Add/Subtract (Left to Right): 40 – 18 = 22
Problem: – 2 × 3²
Solution: – 2 × 9
40 – 18 22
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34. Don’t Give Up! Think of PEMDAS! Parentheses Exponents Multiply/Divide
Add/Subtract
Don’t Give Up!
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35. Additional Resources and Tutorials
– Instruction as well as practice problems are provided for extra help
– Test your knowledge of the steps that you need to follow to simplify an expression
– For those that are up for a challenge, see if you can arrange these numbers to get the answer shown using the order of operations
– Click on “Algebra” then, “Number Operations” and follow along with an interactive tutorial of the order of operations
– Read about and practice using the order of operations to simplify expressions
– Test your skills with several of these different games
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