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# 1.4: Order of Operations

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Aviary Dilemma

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+3×52+3\begin{align*}256 + 3 \times 5 - 2 + 3\end{align*} 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 lesson, you will know all about the order of operations. Then you will be able to help Keisha with the bird count.

What You Will Learn

In this lesson you will learn the following skills.

• 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

Teaching Time

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 "equals" 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\begin{align*}3 + 4 = 7\end{align*}

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\begin{align*}4 + 3 \times 5 \end{align*}

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 to our example.

Example

4+3×5\begin{align*}4 + 3 \times 5\end{align*}

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+3×54+15=19\begin{align*}& 4 + 3 \times 5\\ & 4 + 15\\ & = {19}\end{align*}

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\begin{align*}4 + 3 \times 5\end{align*}

We probably would have solved the problem in order from left to right.

4+3×57×5=35\begin{align*}& 4 + 3 \times 5\\ & 7 \times 5\\ & = 35 \end{align*}

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.

1. 81×4+3=\begin{align*} 8 - 1 \times 4 + 3 = \underline{\;\;\;\;\;\;\;}\end{align*}
2. 2×6+8÷2=\begin{align*}2 \times 6 + 8 \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}
3. 5+9×36+2=\begin{align*}5 + 9 \times 3 - 6 + 2 =\underline{\;\;\;\;\;\;\;}\end{align*}

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 parentheses fall in the order of operations.

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

Wow! You can see that, according to the order of operations, parentheses 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+(31)×2\begin{align*}2 + (3 - 1) \times 2\end{align*}

In this example, we can see that we have four things to look at.

We have 1 set of parentheses, addition, subtraction in the parentheses and multiplication.

We can evaluate this expression using the order of operations.

Example

2+(31)×22+2×22+4=6\begin{align*}& 2 + (3 - 1) \times 2\\ & 2 + 2 \times 2\\ & 2 + 4\\ & = 6\end{align*}

Our answer is 6.

What about when we have parentheses and exponents?

Example

35+32(3×2)×7\begin{align*}35 + 3^2 - (3 \times 2) \times 7\end{align*}

We start by using the order of operations. It says we evaluate parentheses first.

3×2=635+326×7\begin{align*}& 3 \times 2 = 6\\ & 35 + 3^2 - 6 \times 7\end{align*}

Next we evaluate exponents.

32=3×3=935+96×7\begin{align*}& 3^2 = 3 \times 3 = 9\\ & 35 + 9 - 6 \times 7\end{align*}

Next, we complete multiplication or division in order from left to right. We have multiplication.

6×7=4235+942\begin{align*}& 6 \times 7 = 42\\ & 35 + 9 - 42\end{align*}

Next, we complete addition and/or subtraction in order from left to right.

35+94442=44=2\begin{align*}35 + 9 & = 44\\ 44 - 42 & = 2\end{align*}

Our answer is 2. Here are a few for you to try on your own.

1. 16+235+(3×4)\begin{align*}16 + 2^3 - 5 + (3 \times 4)\end{align*}
2. 92+225×(2+3)\begin{align*}9^2 + 2^2 - 5 \times (2 + 3)\end{align*}
3. 82÷2+41×6\begin{align*}8^2 \div 2 + 4 - 1 \times 6\end{align*}

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.

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\begin{align*}4 + 1\end{align*} which is 5. Then we multiply 7×5\begin{align*}7 \times 5\end{align*} and 7×2\begin{align*}7 \times 2\end{align*}. Since multiplication comes next in our order of operations. Finally we subtract 3514=21.\begin{align*}35 - 14 = 21.\end{align*}

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\begin{align*}3 \times 2\end{align*} 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\begin{align*}3 \times 2\end{align*} 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\begin{align*}8 \times 2 = 16\end{align*} and 5×2=10\begin{align*}5 \times 2 = 10\end{align*}. 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\begin{align*}6 \times 3 = 18\end{align*}. Next, we complete the multiplication 2×3=6\begin{align*}2 \times 3 = 6\end{align*}. 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 182\begin{align*}18 - 2\end{align*} before multiplying. You can’t do that. He needed to multiply 2×3\begin{align*}2 \times 3\end{align*} 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.

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+71=22\begin{align*}5 + 3 \times 2 + 7 - 1 = 22\end{align*}

Now if we just solve this problem without parentheses, we get the following answer.

5+3×2+71=17\begin{align*}5 + 3 \times 2 + 7 - 1 = 17\end{align*}

How did we get this answer?

Well, we began by completing the multiplication, 3×2=6\begin{align*}3 \times 2 = 6\end{align*}. 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\begin{align*}5 + 3\end{align*}.

Example

(5+3)×2+71=22\begin{align*}(5 + 3) \times 2 + 7 - 1 = 22\end{align*}

Is this a true statement?

Well, we begin by completing the addition in parentheses, 5+3=8\begin{align*}5 + 3 = 8\end{align*}. Next we complete the multiplication, 8×2=16\begin{align*}8 \times 2 = 16\end{align*}.

Here is our problem now.

16+71=22\begin{align*}16 + 7 - 1 = 22\end{align*}

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. 63+4×2+7=39\begin{align*}6 - 3 + 4 \times 2 + 7 = 39\end{align*}
2. 8×7+3×85=65\begin{align*}8 \times 7 + 3 \times 8 - 5 = 65\end{align*}
3. 2+5×2+184=28\begin{align*}2 + 5 \times 2 + 18 - 4 = 28\end{align*}

Take a minute and check your work with a peer.

## Real Life Example Completed

The Aviary Dilemma

Let’s look back at Keisha and Ms. Thompson and the bird dilemma at the zoo.

Here is the original problem.

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 rescueyesterday.”

“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+3×52+3\begin{align*}256 + 3 \times 5 - 2 + 3\end{align*} that equals 1296 birds, I think. I’m not sure, that doesn’t seem right.”

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\begin{align*}3 \times 5\end{align*} - 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+3×52+3=\begin{align*}256 + 3 \times 5 - 2 + 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

This is the same equation that Keisha came up with. Let’s look at her math.

Keisha says, “That is 256+3×52+3\begin{align*}256 + 3 \times 5 - 2 + 3\end{align*} 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\begin{align*}3 \times 5\end{align*} BEFORE completing any of the other operations.

Let’s look at that.

256+3×52+3=256+152+3=\begin{align*}256 + 3 \times 5 - 2 + 3 = \underline{\;\;\;\;\;\;\;}\\ 256 + 15 - 2 + 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

Now we can complete the addition and subtraction in order from left to right.

256+152+3=272\begin{align*}256 + 15 - 2 + 3 = 272\end{align*}

The new bird count in the aviary is 272 birds.

## Vocabulary

Here are the vocabulary words that appear in this lesson.

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

Grouping Symbols
Parentheses or brackets. Operations in parentheses are completed first according to the order of operations.

## Technology Integration

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!

## Time to Practice

Directions: Evaluate each expression according to the order of operations.

1. \begin{align*}2 + 3 \times 4 + 7 = \underline{\;\;\;\;\;\;\;}\end{align*}

2. \begin{align*}4 + 5 \times 2 + 9 - 1 = \underline{\;\;\;\;\;\;\;}\end{align*}

3. \begin{align*}6 \times 7 + 2 \times 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

4. \begin{align*}4 \times 5 + 3 \times 1 - 9 = \underline{\;\;\;\;\;\;\;}\end{align*}

5. \begin{align*}5 \times 3 \times 2 + 5 - 1 = \underline{\;\;\;\;\;\;\;}\end{align*}

6. \begin{align*}4 + 7 \times3 + 8 \times2 = \underline{\;\;\;\;\;\;\;}\end{align*}

7. \begin{align*}9 - 3 \times 1 + 4 - 7 = \underline{\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}10 + 3 \times 4 + 2 -8 = \underline{\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}11 \times 3 + 2 \times 4 - 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}6 + 7 \times 8 - 9 \times 2 = \underline{\;\;\;\;\;\;\;}\end{align*}

11. \begin{align*}3 + 4^2 - 5 \times 2 + 9 = \underline{\;\;\;\;\;\;\;}\end{align*}

12. \begin{align*}2^2 + 5 \times 2 + 6^2 - 11 = \underline{\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}3^2 \times 2 + 4 - 9 = \underline{\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}6 + 3 \times 2^2 + 7 - 1 = \underline{\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}7 + 2 \times 4 + 3^2 - 5 = \underline{\;\;\;\;\;\;\;}\end{align*}

16. \begin{align*}3 + (2 + 7) - 3 + 5 = \underline{\;\;\;\;\;\;\;}\end{align*}

17. \begin{align*}2 + (5 - 3) + 7^2 - 11 = \underline{\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}4 \times 2 + (6 - 4) - 9 + 5 = \underline{\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}8^2 - 4 + (9 - 3) + 12 = \underline{\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}7^3 - 100 + (3 + 4) - 9 = \underline{\;\;\;\;\;\;\;}\end{align*}

Directions: Check each answer using order of operations. Write whether the answer is true or false.

21. \begin{align*}4 + 5 \times 2 + 8 - 7 = 15\end{align*}

22. \begin{align*}4 + 3 \times 9 + 6 - 10 = 104\end{align*}

23. \begin{align*}6 + 2^2 \times 4 + 3 \times 6 = 150\end{align*}

24. \begin{align*}3 + 6 \times 3 + 9 \times 7 - 18 = 66\end{align*}

25. \begin{align*}7 \times 2^3 + 4 - 9 \times 3 - 8 = 25\end{align*}

Directions: Insert grouping symbols to make each a true statement.

26. \begin{align*}4 + 5 - 2 + 3 - 2 = 8\end{align*}

27. \begin{align*}2 + 3 \times 2 - 4 = 6\end{align*}

28. \begin{align*}1 + 9 \times 4 \times 3 + 2 - 1 = 110\end{align*}

29. \begin{align*}7 + 4 \times 3 - 5 \times 2 = 23\end{align*}

30. \begin{align*}2^2 + 5 \times 8 - 3 + 4 = 33\end{align*}

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