Numerical Expression
Evaluation with Basic
Operations
Jen Kershaw
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both
in the U.S. and worldwide. Using an open-content, web-based
collaborative model termed the FlexBook®, CK-12 intends to
pioneer the generation and distribution of high-quality educational
content that will serve both as core text as well as provide an
adaptive environment for learning, powered through the FlexBook
Platform®.
Copyright © 2013 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/terms.
Printed: October 11, 2013
AUTHOR
Jen Kershaw
www.ck12.org
Concept 1. Numerical Expression Evaluation with Basic Operations
C ONCEPT
1
Numerical Expression
Evaluation with Basic Operations
Here you’ll learn how to evaluate numerical expressions using the four operations.
You wouldn’t think that an aviary would have mathematics in it, but this aviary has a problem and solving numerical
expressions using the four operations is the way to solve it. Have you ever thought about this?
Keisha loves the birds in the aviary at the city zoo. Her favorite part of the aviary is the bird rescue. Here the zoo
staff rescues injured birds, helps them to heal and then releases them again. Currently, they have 256 birds in the
rescue. Today, Keisha has a special visit planned with Ms. Thompson who is in charge of the bird rescue.
When Keisha arrives, Ms. Thompson is already hard at work. She tells Keisha that there are new baby birds in the
rescue. Three of the birds have each given birth to five baby birds. Keisha can’t help grinning as she walks around.
She can hear the babies chirping. In fact, it sounds like they are everywhere.
“It certainly sounds like a lot more babies,” Keisha says.
“Yes,” Ms. Thompson agrees. “We also released two birds from the rescue yesterday.”
“That is great news,” Keisha says smiling.
“Yes, but we also found three new injured birds. Our population has changed again.”
“I see,” Keisha adds, “That is 256 + 3 × 5 − 2 + 3 that equals 1296 birds, I think. I’m not sure, that doesn’t seem
right.”
Is Keisha’s math correct? How many birds are there now? Can you figure it out? This is a bit of a tricky question.
You will need to learn some new skills to help Keisha determine the number of birds in the aviary.
Pay attention. By the end of the Concept, you will know all about the order of operations. Then you will be
able to help Keisha with the bird count.
Guidance
This Concept 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.
1
www.ck12.org
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 "equals" sign. An equation may also be written as a question, requiring you to
"solve" it in order to make both sides equal.
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.
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 Operation is a way of evaluating expressions. It lets you know what order to complete each operation in.
Order of Operations
P - parentheses
E - exponents
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.
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
4 + 15
= 20
2
www.ck12.org
Concept 1. Numerical Expression Evaluation with Basic Operations
When we evaluate this expression using order of operations, our answer is 20.
What would have happened if we had NOT followed the order of operations?
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 a few for you to try on your own.
Example A
8−1×4+3 =
Solution: 7
Example B
2×6+8÷2 =
Solution: 16
Example C
5+9×3−6+2 =
Solution: 28
Here is the original problem once again. Let’s look back at Keisha and Ms. Thompson and the bird dilemma at the
zoo.
Keisha loves the birds in the aviary at the city zoo. Her favorite part of the aviary is the bird rescue. Here the zoo
staff rescues injured birds, helps them to heal and then releases them again. Currently, they have 256 birds in the
rescue. Today, Keisha has a special visit planned with Ms. Thompson who is in charge of the bird rescue.
When Keisha arrives, Ms. Thompson is already hard at work. She tells Keisha that there are new baby birds in the
rescue. Three of the birds have each given birth to five baby birds. Keisha can’t help grinning as she walks around.
She can hear the babies chirping. In fact, it sounds like they are everywhere.
“It certainly sounds like a lot more babies,” Keisha says.
“Yes,” Ms. Thompson agrees. “We also released two birds from the rescue yesterday.”
“That is great news,” Keisha says smiling.
“Yes, but we also found three new injured birds. Our population has changed again.”
“I see,” Keisha adds, “That is 256 + 3 × 5 − 2 + 3 that equals 1296 birds, I think. I’m not sure, that doesn’t seem
right.”
3
www.ck12.org
We have an equation that Keisha wrote to represent the comings and goings of the birds in the aviary. Before we
figure out if Keisha’s math is correct, let’s underline any important information in the problem. As usual, this has
been done for you in the text. Wow, there is a lot going on. Here is what we have to work with.
256 birds
3 × 5 - three birds each gave birth to five baby birds
1. birds were released
2. injured birds were found.
Since we started with 256 birds, that begins our equation. Then we can add in all of the pieces of the problem.
256 + 3 × 5 − 2 + 3 =
This is the same equation that Keisha came up with. Let’s look at her math. Keisha says, “That is 256 + 3 × 5 − 2 + 3
that equals 1296 birds, I think. I’m not sure, that doesn’t seem right.” It isn’t correct. Keisha forgot to use the order
of operations.
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.
Vocabulary
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 Operations
the 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
Guided Practice
Here is one for you to try on your own.
4
www.ck12.org
Concept 1. Numerical Expression Evaluation with Basic Operations
6 + 8 × 4 − 11 + 6 =
Answer
33
Video Review
MEDIA
Click image to the left for more content.
Khan Academy Introduction to Order of Operations
MEDIA
Click image to the left for more content.
James Sousa Example of Order of Operations
MEDIA
Click image to the left for more content.
James Sousa Example 2 of Order of Operations
Practice
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 =
5
www.ck12.org
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 =
6