1.2: Order of Operations
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 thirteen boys and fourteen 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.
What You Will Learn
By the end of this lesson you will learn how to execute the following skills.
- Evaluate numerical and variable expressions involving arithmetic operations.
- Evaluate numerical and variable expressions involving grouping symbols.
- Write variable expressions to represent and solve real-world problems using order of operations.
Teaching Time
I. Evaluate Numerical and Variable Expressions Involving Arithmetic Operations
In mathematics, you will often hear the word “evaluate.” Before we begin the lesson, it is important for you to understand what the word “evaluate” means. When we evaluate a mathematical sentence, we figure out the value of the number sentence. If the mathematical sentence has numbers in it, then we figure out the value of the number sentence. Often times we think of evaluating as solving, and it can be that, but more specifically, evaluating is figuring out the value of a sentence.
What do we evaluate?
In mathematics, we can evaluate different types of number sentences. Sometimes we will be working with equations and other times we will be working with expressions.
That is a great question. An equation is a number sentence with an equals sign. Therefore, the quantity on one side of the equals sign is equal to the quantity on the other side of the equals sign. An expression is a group of numbers, symbols and variables that represents a quantity; there is not an equal sign. We evaluate an expression to figure out the value of the mathematical statement itself, we are not trying to make one side equal another, as with an equation.
Let’s start by focusing on evaluating expressions. Think about this example.
Example
Two eighth grade math students evaluated the expression: \begin{align*}2 + 3 \times 4 \div 2\end{align*}. Both students approached the expression differently. Macy’s answer was ten. Cole’s answer was eight.
We can think about what happened here. How could one student arrive at one answer and another student come up with an entirely different answer? The key is in the order that each student performed each operation.
This is where the order of operations comes in. The order of operations is a rule that tells us which operation we need to perform in what order to achieve the accurate answer. The order of operations applies whenever you have two or more operations in a single expression. Here is the order of operations.
Order of Operations
P parentheses or grouping symbols
E exponents
MD multiplication and division in order from left to right
AS addition and subtraction in order from left to right
Take a few minutes to write down the order of operations in your notebook.
Now the example above does not have any parentheses or exponents, so don’t worry about those just yet. Let’s look at the example again and look at how Macy and Cole arrived at their answers.
Example
\begin{align*}2 + 3 \times 4 \div 2\end{align*}
Cole worked on evaluating this expression using the order of operations. He multiplied \begin{align*}3 \times 4 = 12\end{align*}. Then he divided by 2 which is 6 and finally he added 2 for a final answer of 8. This is correct. It may seem out of order to work this way, but remember you are working according to the order of operations.
What did Macy do? Macy worked on evaluating the expression by working in order from left to right. She simply did not follow the order of operations. In this case, she evaluated the value of the expression as 10. This is incorrect. Next time, Macy needs to follow the order of operations.
Working in this way is called evaluating a numerical expression. It is a numerical expression because it is made up only of numbers and operations.
Example
Evaluate \begin{align*}3 + 9 \cdot 2 \div 3 + 8\end{align*}
To begin with, we first need to remind ourselves of the order of operations. Notice that there is a dot in between the nine and the two. This is another way to show multiplication. As you get into higher levels of math, you will see that multiplication is often shown in other ways besides using an \begin{align*}x\end{align*}. Now back to evaluating.
Following the order of operations, we would first multiply and then divide.
\begin{align*}9 \cdot 2 = 18\end{align*}
\begin{align*}18 \div 3 = 6\end{align*}
Next we perform addition and subtraction in order from left to right.
\begin{align*}6 + 3 = 9\end{align*}
\begin{align*}9 + 8 = 17\end{align*}
This is our answer.
You will also find another type of expression. These expressions can have letters in them. These letters are called variables and variables represent an unknown quantity. When you see an expression with a variable in it, we call it a variable expression.
We can evaluate variable expressions using the order of operations too. The key thing with a variable expression is that you will have to substitute a given value into the expression for the unknown variable and then evaluate the expression. Let’s look at an example.
Example
Evaluate the expression \begin{align*}60 \div 2 \cdot 2a + 16 - 4\end{align*} when \begin{align*}a = 5\end{align*}.
First, notice that the expression has the letter \begin{align*}a\end{align*} in it. This is our variable. Also, you can see that you have been given a value for \begin{align*}a\end{align*}. Our first step is to substitute the value of \begin{align*}a\end{align*} into the expression.
\begin{align*}60 \div 2 \cdot 2(5) + 16 - 4\end{align*}
Another good question-using the parentheses is another way to show multiplication. So now you have three ways to show multiplication. You can use an \begin{align*}x\end{align*}, a dot or a set of parentheses around a number. This means that we are multiplying the 2 times the 5.
Now we can use our order of operations. We have division, multiplication and multiplication in this expression right away. We complete multiplication and division in order from left to right.
Another good question! In this case, the set of parentheses is not grouping two numbers and an operation. When talking about parentheses in the order of operations, we need to have an operation inside it. The set of parentheses in this example is being used to show multiplication. There isn’t multiplication inside the parentheses.
Now back to evaluating the expression by performing multiplication and division in order from left to right.
\begin{align*}& 60 \div 2 \cdot 2(5) + 16 - 4\\ & 60 \div 2 = 30\\ & 30 \cdot 2(5) = 300\end{align*}
Next, we work with addition and subtraction in order from left to right.
\begin{align*}& 300 + 16 - 4\\ & 316 - 4\\ & 312\end{align*}
Our final answer is 312.
Example
Evaluate the expression \begin{align*}14 \cdot 2 \div 7 + 3b - 4\end{align*} when \begin{align*}b = 12\end{align*}.
First, we substitute the value of \begin{align*}b\end{align*} into our variable expression.
\begin{align*}14 \cdot 2 \div 7 + 3(12) - 4\end{align*}
Now we follow the order of operations by performing multiplication and division in order from left to right.
\begin{align*}14 \cdot 2 = 28\end{align*}
\begin{align*}28 \div 7 = 4\end{align*}
Let’s rewrite what we have so far so we don’t get confused.
\begin{align*}4 + 3(12) - 4\end{align*}
OH! There’s more multiplication to do!
\begin{align*}3(12) = 36\end{align*}
Now our expression is:
\begin{align*}4 + 36 - 4\end{align*}
Our last step is to perform addition and subtraction in order from left to right.
\begin{align*}4 + 36 = 40 - 4 = 36\end{align*}
Our final answer is 36.
Now that you have had some practice the order of operations is probably beginning to make more sense. If you always follow them, then your work will be accurate. You can think of the order of operations as a kind of road map for working with complicated expressions.
Next, we are going to look at the first step in the order of operations. Let’s talk about grouping symbols.
II. Evaluate Numerical and Variable Expressions Involving Grouping Symbols
Grouping symbols are a way to point out an operation that needs special attention. If you can think of a spotlight, grouping symbols are a way of highlighting or shining a spotlight on a particular operation.
What are the grouping symbols?
The grouping symbols that we are going to be working with are brackets [ ] and parentheses ( ). According to the order of operations, we perform all operations inside of grouping symbols BEFORE performing any other operation in the list.
Example
Evaluate the expression \begin{align*}7 + 4(15 \div 5) - 6\end{align*}.
First, notice that we have a set of parentheses in this numerical expression. Remember that it is called a numerical expression because this expression does not have any variable in it.
We perform any and all operations in parentheses first.
\begin{align*}15 \div 5 = 3\end{align*}
Now let’s rewrite the expression.
\begin{align*}7 + 4(3) - 6\end{align*}
Our next step is to continue with the order of operations. We have multiplication in this expression. We do that next.
\begin{align*}7 + 12 - 6\end{align*}
Now we can perform addition and subtraction in order from left to right.
\begin{align*}19 - 6 = 13\end{align*}
Our final answer is 13.
We can also evaluate variable expressions that contain parentheses.
Example
Evaluate the expression \begin{align*}6y + 3 - (2 \cdot 4)\end{align*} when \begin{align*}y = 15\end{align*}.
First, substitute the value in for the unknown quantity, \begin{align*}y\end{align*}.
\begin{align*}6(15) + 3 - (2 \cdot 4)\end{align*}
Our next step is to perform the operation in parentheses.
\begin{align*}6(15) + 3 - 8\end{align*}
Now we can perform multiplication and division in order from left to right.
\begin{align*}90 + 3 - 8\end{align*}
Finally, we work with addition and subtraction in order from left to right.
\begin{align*}& 93 - 8 \\ & 85\end{align*}
Our answer is 85.
What about brackets?
Brackets can be used to group more than one operation. When you see a set of brackets, remember that brackets are a way of grouping numbers and operations. There is a spotlight on brackets too.
Example
Evaluate the expression \begin{align*}6 + [5 + (4 \cdot 6)] - 17\end{align*}.
Now we have a set of parentheses within a set of brackets. To work through this, we are going to perform the operation in the parentheses inside the brackets first, and then we will perform the other operation in the brackets.
\begin{align*}& 6 + [5 + 24] - 17\\ & 6 + 29 - 17\end{align*}
Next we perform addition and subtraction in order from left to right.
\begin{align*}& 35 - 17\\ & 18\end{align*}
Our final answer is 18.
Remember that when you are following the road map laid out by the order of operations that you need to include grouping symbols of brackets and parentheses in that order!
III. Write Variable Expressions to Represent and Solve Real-World Problems Using Order of Operations
When we have an unknown quantity in a problem, we can use a variable to help us to find the value of our problem. You can find many real-world scenarios where variable expressions would be helpful in solving problems.
Let’s think about ticket pricing. Many different places sell tickets. There are also prices for adults and prices for children. The number of tickets can vary, or they can be the same. If we know the number of tickets, then we can figure out the total amount of revenue earned based on the cost of the ticket for an adult and for a child. Let’s look at an example to see how this works.
Example
An amusement park charges eight dollars admission and one dollar and fifty-cents per ride. Write an expression to find the cost of admission and five ride tickets.
First, notice that we have eight dollars admission. That is the first part of the expression.
8
Next, they charge $1.50 per ride.
\begin{align*}8 + 1.50x\end{align*}
We used the variable for the number of rides since this is the variable or changeable facet. The number of rides can change. In this example, we were asked to figure out the cost for 5 rides. That is the value that we can substitute into our expression for \begin{align*}x\end{align*}.
\begin{align*}8 + 1.50(5)\end{align*}
Now we can use the order of operations to solve this problem.
\begin{align*}& 8 + 7.50 \\ & 15.50\end{align*}
The cost of admission plus five ride tickets is $15.50.
Using variable expressions can help you solve for unknown quantities. Just remember to use the order of operations so that your work is accurate!!
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 thirteen boys and fourteen 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.
\begin{align*}11 + 13\end{align*}
\begin{align*}14 + 14\end{align*}
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.
\begin{align*}(11 + 13) \div 4\end{align*}
Or \begin{align*}\frac{11+13}{4}\end{align*}
This is an expression for the boys.
\begin{align*}(14 + 14) \div 4\end{align*}
Or \begin{align*}\frac{14+14}{4}\end{align*}
This is an expression for the girls.
Now we can solve for the number on each team.
\begin{align*}\text{Boys} = (11 + 13) \div 4 = 24 \div 4 = 6\end{align*} boys on each team
\begin{align*}\text{Girls} = (14 + 14) \div 4 = 28 \div 4 = 7\end{align*} girls on each team
Now our work is complete.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Evaluate
- to figure out the value of a numerical or variable expression.
- Equation
- a mathematical statement with an equals sign where one side of the equation has the same value as the other side.
- Expression
- a group of numbers, symbols and variables that represents a quantity.
- Numerical Expression
- a group of numbers and operations.
- Variable Expression
- a group of numbers, operations and at least one variable.
- Variable
- a letter used to represent an unknown quantity.
- Grouping Symbols
- parentheses or brackets used to group numbers and operations.
Time to Practice
Directions: Evaluate each numerical expression using the order of operations.
- \begin{align*}4 + 5 \cdot 2 - 3\end{align*}
- \begin{align*}6 + 6 \cdot 3 \div 2 - 7\end{align*}
- \begin{align*}5 + 5 \cdot 8 \div 2 + 6\end{align*}
- \begin{align*}13 - 3 \cdot 2 + 8 - 2\end{align*}
- \begin{align*}17 - 5 \cdot 3 + 8 \div 2\end{align*}
- \begin{align*}9 + 4 \cdot 2 + 7 - 1\end{align*}
- \begin{align*}8 + 5 \cdot 6 + 2 \cdot 4 - 3\end{align*}
- \begin{align*}19 + 2 \cdot 4 - 3 \cdot 2 + 10\end{align*}
- \begin{align*}12 + 4 \cdot 4 \div 8 - 3\end{align*}
- \begin{align*}12 \cdot 2 + 16 \div 2 - 12\end{align*}
Directions: Evaluate each variable expression. Remember to use the order of operations when necessary.
- \begin{align*}4y+6-2\end{align*}, when \begin{align*}y=6\end{align*}
- \begin{align*}9+3x-5+2\end{align*}, when \begin{align*}x=8\end{align*}
- \begin{align*}6y+2y-5\end{align*}, when \begin{align*}y=3\end{align*}
- \begin{align*}8+3y-5 \cdot 2\end{align*}, when \begin{align*}y=4\end{align*}
- \begin{align*}7x-2 \cdot 3 \div 3+12\end{align*}, when \begin{align*}x=5\end{align*}
- \begin{align*}3+4 \cdot 3 - 2y+5\end{align*}, when \begin{align*}y=7\end{align*}
- \begin{align*}6a+3(2)+5 - 4\end{align*}, when \begin{align*}a=9\end{align*}
- \begin{align*}10+3 \cdot 5+2-9b\end{align*}, when \begin{align*}b=2\end{align*}
- \begin{align*}14 \div 2+3a+7a\end{align*}, when \begin{align*}a=2\end{align*}
- \begin{align*}5+6y-2y+11-4\end{align*}, when \begin{align*}y=3\end{align*}
Directions: Evaluate each expression using the order of operations. Remember to pay attention to the grouping symbols.
- \begin{align*}3 + (4 + 5) - 6(2)\end{align*}
- \begin{align*}4 + (6 \div 3) + 2(7) - 4\end{align*}
- \begin{align*}3 + 2(4 + 2) - 5(2)\end{align*}
- \begin{align*}7 + (3 + 2) - 5 + 8(3)\end{align*}
- \begin{align*}4(2) + (3 + 9) - 4\end{align*}
- \begin{align*}7 + [4 + (3 \cdot 2)] - 5\end{align*}
- \begin{align*}9 + [6 - (2 \cdot 3)] + 15\end{align*}
- \begin{align*}4 \cdot 2 + [3 + (7 + 2)] - 4\end{align*}