3.2 Order of Operations

Learning Objectives

You will learn how to evaluate numerical expressions using the order of operations involving arithmetic operations as well as grouping symbols.

Introduction

Gym Class Changes

Wood shop wasn’t the only thing that had changed with the new year. After working to successfully resolve wood shop being reinstated, the students found out that there were also changes for gym class or physical education. It seems that the old gym teacher, Mr. Woullard had retired and there was a new gym teacher.

Mr. Osgrove was young and lively with lots of energy, but he also had some new ideas about how gym class ought to be run.

“We’ll combine two periods of students together,” he explained. “That will give us so many more combinations of students when it comes to teams.”

Jesse looked around. He counted the number of boys and girls in his class. There were 11 boys and 14 girls in the class. The other class had 13 boys and 14 girls in it.

“We can add the boys together and form four teams and the girls together and form four teams.”

Jesse left gym class with his head full of numbers. If they were to combine all of the boys from the two classes and all of the girls from the two classes, then that would be a lot of students. Jesses started to figure out the different combinations of teams.

To work this through, you can use the order of operations. That is what you will learn in this lesson. Pay close attention, then you can work this through at the end of the lesson.

Guided Learning

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.

For 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 A

$4 + 3 \times 5$

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.

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.

$4 + 3 \times 5$

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.

$& 4 + 3 \times 5\\& 4 + 15\\& = {20}$

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?

$4 + 3 \times 5$

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

$& 4 + 3 \times 5\\& 7 \times 5\\& = 35$

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:

$8 - 1 \times 4 + 3 = \underline{\;\;\;\;\;\;\;}$
$2 \times 6 + 8 \div 2 = \underline{\;\;\;\;\;\;\;}$
$5 + 9 \times 3 - 6 + 2 =\underline{\;\;\;\;\;\;\;}$

Take a few minutes and check your work with a peer.

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 B

$2 + (3 - 1) \times 2$

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.

$& 2 + (3 - 1) \times 2\\& 2 + 2 \times 2\\& 2 + 4\\& = 6$

Our answer is 6.

What about when we have parentheses and exponents?

Example C

$35 + 3^2 - (3 \times 2) \times 7$

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

$& 3 \times 2 = 6\\& 35 + 3^2 - 6 \times 7$

Then, we evaluate exponents.

$& 3^2 = 3 \times 3 = 9\\& 35 + 9 - 6 \times 7$

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

$& 6 \times 7 = 42\\& 35 + 9 - 42$

Finally, 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.

$16 + 2^3 - 5 + (3 \times 4)$
$9^2 + 2^2 - 5 \times (2 + 3)$
$8^2 \div 2 + 4 - 1 \times 6$

Take a minute and check your work with a peer.

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 which is 5. Then we multiply $7 \times 5$ and $7 \times 2$ . 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 $3 \times 2$ 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 $3 \times 2$ first, then you get 19 as an answer instead of 16.

Problem Number 4:

Let’s complete the work in parentheses first, $8 \times 2 = 16$ and $5 \times 2 = 10$ . 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, $6 \times 3 = 18$ . Next, we complete the multiplication $2 \times 3 = 6$ . 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 $2 \times 3$ 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.

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 D

$5 + 3 \times 2 + 7 - 1 = 22$

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

$5 + 3 \times 2 + 7 - 1 = 17$

How did we get this answer?

Well, we began by completing the multiplication, $3 \times 2 = 6$ . 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 .

$(5 + 3) \times 2 + 7 - 1 = 22$

Is this a true statement?

Well, we begin by completing the addition in parentheses, 5 + 3 = 8 . Next we complete the multiplication, $8 \times 2 = 16$ .

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.

$6 - 3 + 4 \times 2 + 7 = 39$
$8 \times 7 + 3 \times 8 - 5 = 65$
$2 + 5 \times 2 + 18 - 4 = 28$

Take a minute and check your work with a peer.

Real Life Example Completed

Gym Class Changes

Here is the original problem once again. Reread it and then write two expressions to show how the teams will be divided up. How many boys will be on each team if there are four teams? How many girls will be on four teams? There are four parts to your answer.

Wood shop wasn’t the only thing that had changed with the new year. After working to successfully resolve wood shop being reinstated, the students found out that there were also changes for gym class or physical education. It seems that the old gym teacher, Mr. Woullard had retired and there was a new gym teacher.

Mr. Osgrove was young and lively with lots of energy, but he also had some new ideas about how gym class ought to be run.

“We’ll combine two periods of students together,” he explained. “That will give us so many more combinations of students when it comes to teams.”

Jesse looked around. He counted the number of boys and girls in his class. There were 11 boys and 14 girls in the class. The other class had 13 boys and 14 girls in it.

“We can add the boys together and form four teams and the girls together and form four teams.”

Jesse left gym class with his head full of numbers. If they were to combine all of the boys from the two classes and all of the girls from the two classes, then that would be a lot of students. Jesses started to figure out the different combinations of teams.

Now it is time to work on this solution. Write two expressions to show how the groups are divided. Then answer the two questions of how many boys will be on a team and how many girls will be on a team.

Solution to Real – Life Example

First, let’s look at the information that we have been given in the problem.

Class one has 11 boys and 14 girls.

Class two has 13 boys and 14 girls.

The boys from the two classes will be added together, and the girls from the two classes will be added together.

We can use parentheses to show that the boys will be added and the girls will be added. Both groups will be divided by four.

$(11 + 13) \div 4$

Or $\frac{11+13}{4}$

This is an expression for the boys.

$(14 + 14) \div 4$

Or $\frac{14+14}{4}$

This is an expression for the girls.

Now we can solve for the number on each team.

$\text{Boys} = (11 + 13) \div 4 = 24 \div 4 = 6$ boys on each team

$\text{Girls} = (14 + 14) \div 4 = 28 \div 4 = 7$ girls on each team

Now our work is complete.

Review

As you use the order of operations, make sure to use addition and subtraction in order from left to right if those are the only two operations remaining.
As you use the order of operations, make sure to use multiplication and division in order from left to right if those are the only two operations remaining.

Equation

A number sentence that describes two values that are the same, or equal, to each other.

Order of Operations

The way of evaluating expressions with certain operations being completed first.

Video Resources

Khan Academy Introduction to Order of Operations

James Sousa Example of Order of Operations

http://www.teachertube.com/members/viewVideo.php?video_id=11148

http://www.schooltube.com/video/b828ac92b85e45478188/