1.4: Evaluate Numerical and Variable Expressions Using the Order of Operations
The students at Washington Middle School experienced changes with their new school year. Their old gym teacher, Mr. Woullard had retired and there was a new gym teacher who greeted them on the first day of eighth grade.
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 eleven boys and fourteen 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 we can add 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. Jesse began figuring out the different combinations of teams. Jesse knows that he will need to use the order of operations.
Did you know that? Pay attention and you will be able to help Jesse at the end of the Concept.
Guidance
In mathematics, you will often hear the word “evaluate.” Before we begin, 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.
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.
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.
Let’s look at how Macy and Cole arrived at their answers.
\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.
Here is another one.
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.
Take a look.
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 problem 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.
Now let's add in 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.
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.
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.
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.
Example A
Evaluate the expression \begin{align*}6y + 3 - (2 \cdot 4)\end{align*} when \begin{align*}y = 15\end{align*}.
Solution: \begin{align*}85\end{align*}
Example B
\begin{align*}7 + [4 + (3 \cdot 2)] - 5\end{align*}
Solution: \begin{align*}12\end{align*}
Example C
\begin{align*}9 + [6 - (2 \cdot 3)] + 15\end{align*}
Solution: \begin{align*}24\end{align*}
Now back to the dilemma from the beginning of the Concept. 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
- 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.
Guided Practice
Here is one for you to try on your own.
Evaluate the expression \begin{align*}14 \cdot 2 \div 7 + 3b - 4\end{align*} when \begin{align*}b = 12\end{align*}.
Solution
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.
Video Review
Khan Academy Order of Operations
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*}
Brackets
Brackets [ ], are symbols that are used to group numbers in mathematics. Brackets are the 'second level' of grouping symbols, used to enclose items already in parentheses.Equation
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.Evaluate
To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.Expression
An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.Grouping Symbols
Grouping symbols are parentheses or brackets used to group numbers and operations.nested parentheses
Nested parentheses describe groups of terms inside of other groups. By convention, nested parentheses may be identified with other grouping symbols, such as the braces "{}" and brackets "[]" in the expression . Always evaluate parentheses from the innermost set outward.Numerical expression
A numerical expression is a group of numbers and operations used to represent a quantity.Parentheses
Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.PEMDAS
PEMDAS (Please Excuse My Daring Aunt Sally) is a mnemonic device used to help remember the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.Real Number
A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.Variable Expression
A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.Image Attributions
Here you'll learn to evaluate numerical and variable expressions using the order of operations.