Order of Operations
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Printed: September 14, 2015
CONTRIBUTORS
Sarah Klocke
Mary Kurvers
Tom Youngblom
www.ck12.org
C HAPTER
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Chapter 1. Order of Operations
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Order of Operations
Evaluating numerical expressions involving the four arithmetic operations
Evaluating numerical expressions involving powers and grouping symbols
Using the order of operations to determine if an answer is true
Inserting grouping symbols to make a given answer true
Writing numerical expressions to represent real-world problems and solving them using the order of operations
Introduction
Guided Learning
I. Evaluating Numerical Expressions with the Four Arithmetic Operations
This lesson begins with evaluating numerical expressions. Before we can do that we need to answer one key question,
“What is an expression ?”
To understand what an expression is, let’s compare it with an equation.
An equation is a number sentence that describes two values that are the same, or equal, to each other. The
values are separated by the "equal" sign. An equation may also be written as a question, requiring you to
"solve" it in order to make both sides equal.
Example
3+4 = 7
This is an equation. It describes two equal quantities, 3 + 4, and 7.
What is an expression then?
An expression is a number sentence without an equals sign. It can be simplified and/or evaluated.
Example
4+3×5
This kind of expression can be confusing because it has both addition and multiplication in it.
Do we need to add or multiply first?
To figure this out, we are going to learn something called the Order of Operations.
The Order of Operations is a way of evaluating expressions. It lets you know what order to complete each
operation in.
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Order of Operations
G - grouping
ER - exponents or roots
MD - multiplication or division in order from left to right
AS - addition or subtraction in order from left to right
Take a few minutes to write these down in a notebook.
Now that you know the order of operations, let’s go back to our example.
Example
4+3×5
Here we have an expression with addition and multiplication.
We can look at the order of operations and see that multiplication comes before addition. We need to complete that
operation first.
4+3×5
3 × 5 = 15
4 + 15 = 19
When we evaluate this expression using order of operations, our answer is 19.
What would have happened if we had NOT followed the order of operations?
Example
4+3×5
We probably would have solved the problem in order from left to right.
4+3×5
7×5
= 35
This would have given us an incorrect answer. It is important to always follow the order of operations.
Here are few for you to try on your own.
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Chapter 1. Order of Operations
1. 8 − 1 × 4 + 3 =
2. 2 × 6 + 8 ÷ 2 =
3. 5 + 9 × 3 − 6 + 2 =
Take a few minutes and check your work with a peer.
II. Evaluating Numerical Expressions Using Powers and Grouping Symbols
We can also use the order of operations when we have exponent powers and grouping symbols like parentheses.
In our first section, we didn’t have any expressions with exponents or parentheses.
In this section, we will be working with them too.
Let’s review where exponents and grouping fall in the order of operations.
Order of Operations
G - grouping
ER - exponents or roots
MD - multiplication or division in order from left to right
AS - addition or subtraction in order from left to right
Wow! You can see that, according to the order of operations, groupings come first. We always do the work in
parentheses first. Then we evaluate exponents.
Let’s see how this works with a new example.
Example
2 + (3 − 1) × 2
In this example, we can see that we have four things to look at. We have one set of parentheses, subtraction (in
parentheses), addition and multiplication.
We can evaluate this expression using the order of operations.
Example
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2 + (3 − 1) × 2
(3 − 1) = 2
2+2×2
2×2 = 4
2+4
=6
Our answer is 6.
What about when we have parentheses and exponents?
Example
35 + 32 − (3 × 2) × 7
We start by using the order of operations. It says we evaluate parentheses first.
3×2 = 6
35 + 32 − 6 × 7
Next we evaluate exponents.
32 = 3 × 3 = 9
35 + 9 − 6 × 7
Next, we complete multiplication or division in order from left to right. We have multiplication.
6 × 7 = 42
35 + 9 − 42
Next, we complete addition and/or subtraction in order from left to right.
35 + 9 = 44
44 − 42 = 2
Our answer is 2. Here are a few for you to try on your own.
1. 16 + 23 − 5 + (3 × 4)
2. 92 + 22 − 5 × (2 + 3)
3. 82 ÷ 2 + 4 − 1 × 6
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Chapter 1. Order of Operations
Take a minute and check your work with a peer.
III. Use the Order of Operations to Determine if an Answer is True
We just finished using the order of operations to evaluate different expressions.
We can also use the order of operations to “check” our work.
In this section, you will get to be a “Math Detective.”
As a math detective, you will be using the order of operations to determine whether or not someone else’s work is
correct.
Here is a worksheet that has been completed by Joaquin.
Your task is to check Joaquin’s work and determine whether or not his work is correct.
Use your notebook to take notes.
If the expression has been evaluated correctly, then please make a note of it. If it is incorrect, then re-evaluate the
expression correctly.
Here are the problems that are on Joaquin’s worksheet.
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Did you check Joaquin’s work?
Let’s see how you did with your answers. Take your notebook and check your work with these correct answers.
Let’s begin with problem number 1.
We start by adding 4 + 1 which is 5. Then we multiply 7 × 5 and 7 × 2. Since multiplication comes next in our order
of operations. Finally we subtract 35 − 14 = 21.
Joaquin’s work is correct.
Problem Number 2
We start by evaluating the parentheses. 3 times 2 is 6. Next, consider the exponents. 3 squared is 9 and 4 squared is
16. Finally we can complete the addition and subtraction in order from left to right. Our final answer is 22. Joaquin’s
work is correct.
Problem Number 3
We start with the parentheses, and find that 7 minus 1 is 6. There are no exponents to evaluate, so we can move to
the multiplication step. Multiply 3 × 2 which is 6. Now we can complete the addition and subtraction in order from
left to right. The answer correct is 13. Uh Oh, Joaquin’s answer is incorrect. How did Joaquin get 19 as an answer?
Well, if you look, Joaquin did not follow the order of operations. He just did the operations in order from left to
right. If you don’t multiply 3 × 2 first, then you get 19 as an answer instead of 16.
Problem Number 4
Let’s complete the work in parentheses first, 8 × 2 = 16 and 5 × 2 = 10. Next we evaluate the exponent, 3 squared is
9. Now we can complete the addition and subtraction in order from left to right. The answer is 17.
Joaquin’s work is correct.
Problem Number 5
First, we need to complete the work in parentheses, 6×3 = 18. Next, we complete the multiplication 2×3 = 6. Now,
we can evaluate the addition and subtraction in order from left to right. Our answer is 30.
Uh Oh, Joaquin got mixed up again. How did he get 66? Let’s look at the problem. Oh, Joaquin subtracted 18 − 2
before multiplying. You can’t do that. He needed to multiply 2 × 3 first then he needed to subtract. Because of this,
Joaquin’s work is not accurate.
How did you do?
Remember, a Math Detective can check any answer by following the order of operations.
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Chapter 1. Order of Operations
IV. Insert Grouping Symbols to Make a Given Answer True
Sometimes a grouping symbol can help us to make an answer true. By putting a grouping symbol, like parentheses,
in the correct spot, we can change an answer.
Let’s try this out.
Example
5 + 3 × 2 + 7 − 1 = 22
Now if we just solve this problem without parentheses, we get the following answer.
5 + 3 × 2 + 7 − 1 = 17
How did we get this answer?
Well, we began by completing the multiplication, 3 × 2 = 6. Then we completed the addition and subtraction in
order from left to right. That gives us an answer of 17.
However, we want an answer of 22.
Where can we put the parentheses so that our answer is 22?
This can take a little practice and you may have to try more than one spot too.
Let’s try to put the parentheses around 5 + 3.
Example
(5 + 3) × 2 + 7 − 1 = 22
Is this a true statement?
Well, we begin by completing the addition in parentheses, 5+3 = 8. Next we complete the multiplication, 8×2 = 16.
Here is our problem now.
16 + 7 − 1 = 22
Next, we complete the addition and subtraction in order from left to right.
Our answer is 22.
Here are a few for you to try on your own. Insert a set of parentheses to make each a true statement.
1. 6 − 3 + 4 × 2 + 7 = 39
2. 8 × 7 + 3 × 8 − 5 = 65
3. 2 + 5 × 2 + 18 − 4 = 28
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Take a minute and check your work with a peer.
According to the order of operations, Keisha needed to multiply 3 × 5 BEFORE completing any of the other
operations.
Let’s look at that.
256 + 3 × 5 − 2 + 3 =
256 + 15 − 2 + 3 =
Now we can complete the addition and subtraction in order from left to right.
256 + 15 − 2 + 3 = 272
The new bird count in the aviary is 272 birds.
Technology Integration
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/5257
Khan Academy Introduction to Order of Operations
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/5258
James Sousa Example of Order of Operations
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/5259
James Sousa Example of Order of Operations
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Chapter 1. Order of Operations
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/5260
James Sousa Example of Order of Operations
Here are some additional videos that present Order of Operations in a creative way.
1. http://www.teachertube.com/members/viewVideo.php?video_id=11148 - You will need to register with this
website. This is a fantastic video of a creative teacher teaching the order of operations in a kinesthetic way.
2. http://www.schooltube.com/video/b828ac92b85e45478188/ - This is the Pemdas Parrot song! Very fun and
creative!
Practice Set
Directions: Evaluate each expression according to the order of operations.
1. 2 + 3 × 4 + 7 =
2. 4 + 5 × 2 + 9 − 1 =
3. 6 × 7 + 2 × 3 =
4. 4 × 5 + 3 × 1 − 9 =
5. 5 × 3 × 2 + 5 − 1 =
6. 4 + 7 × 3 + 8 × 2 =
7. 9 − 3 × 1 + 4 − 7 =
8. 10 + 3 × 4 + 2 − 8 =
9. 11 × 3 + 2 × 4 − 3 =
10. 6 + 7 × 8 − 9 × 2 =
11. 3 + 42 − 5 × 2 + 9 =
12. 22 + 5 × 2 + 62 − 11 =
13. 32 × 2 + 4 − 9 =
14. 6 + 3 × 22 + 7 − 1 =
15. 7 + 2 × 4 + 32 − 5 =
16. 3 + (2 + 7) − 3 + 5 =
17. 2 + (5 − 3) + 72 − 11 =
18. 4 × 2 + (6 − 4) − 9 + 5 =
19. 82 − 4 + (9 − 3) + 12 =
20. 73 − 100 + (3 + 4) − 9 =
Directions: Check each answer using order of operations. Write whether the answer is true or false.
21. 4 + 5 × 2 + 8 − 7 = 15
22. 4 + 3 × 9 + 6 − 10 = 104
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23. 6 + 22 × 4 + 3 × 6 = 150
24. 3 + 6 × 3 + 9 × 7 − 18 = 66
25. 7 × 23 + 4 − 9 × 3 − 8 = 25
Directions: Insert grouping symbols to make each a true statement.
26. 4 + 5 − 2 + 3 − 2 = 8
27. 2 + 3 × 2 − 4 = 6
28. 1 + 9 × 4 × 3 + 2 − 1 = 110
29. 7 + 4 × 3 − 5 × 2 = 23
30. 22 + 5 × 8 − 3 + 4 = 33
Review
Follow the Order of Operations to correctly solve expressions and equations.
Expression
a number sentence with operations and no equals sign.
Equation
a number sentence that compares two quantities that are the same. It has an equals sign in it and may be
written as a question requiring a solution.
Order of Operationsthe order that you perform operations when there is more than one in an expression or equation.
P - parentheses
E - exponents
MD - multiplication/division in order from left to right
AS - addition and subtraction in order from left to right
Grouping Symbols
Parentheses or brackets. Operations in parentheses are completed first according to the order of operations.
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