1.4: Order of Operations
Introduction
Dividing up the Groups
The first few days of Teen Adventure were spent at the base of the White Mountains in the Lafayette Place Campground. There the teens worked on survival skills, map skills, and basic first aid. The students were kept together and Kelly made some terrific new friends. Her favorite part was all of the team building and trust building exercises that they learned. She even surprised herself by being able to complete some tasks that she wasn’t sure that she could do. Kelly and the other students learned to rely on themselves and each other over those first few days.
The night before their first hike, the group leaders told them that they would be divided up into hiking groups. Each week they would meet altogether once again to talk about their adventures, but during the actual week each group would be on a different trail. There would be three groups out of the 30 students. The 6 leaders would be divided into the 3 groups. For the first few days, there would be a manager along and two first-aid persons. After the first few days, the manager and the two first aid persons would head back to the base office to be checked in with weekly or in case of emergency. So they would start with one number of people and after the first few days the number would change.
Kelly is curious about how many people will start and how many will be together after the first few days.
This is where you come in. You can write an expression to figure this out-using grouping symbols and the order of operations you should be able to calculate two different group numbers. This lesson will teach you all that you need to know. Pay attention because you will see this problem once again.
What You Will Learn
In this lesson you will learn to perform the following tasks.
- Evaluate variable expressions involving the four arithmetic operations.
- Evaluate variable expressions involving powers and grouping symbols.
- Write variable expressions to represent real-world problems using order of operations.
Teaching Time
I. Evaluate Variable Expressions Involving the Four Arithmetic Operations
To begin this lesson, you have to think back to last year in math. Think back to last year in math, do you remember using the order of operations to evaluate numerical expressions? Remember that a numerical expression is an expression that has numbers and operations in it. Let’s review with an example.
Example
\begin{align*}2 + 3 \times 5\end{align*}
We could evaluate this expression in two different ways.
The first way would be to simply perform the operations in order from left to right.
\begin{align*}2 + 3 = 5 \times 5 = 25\end{align*}
Wait a minute!! That is incorrect because we did not use the order of operations.
Let’s use the order of operations as we evaluate this expression a second way.
\begin{align*}2 + 3 \times 5\end{align*}
Good, now we can apply this information to our example.
\begin{align*}2 + 3 \times 5\end{align*}
We multiply first.
\begin{align*}3 \times 5 = 15\!\\ 2 + 15 = 17\end{align*}
Our answer is 17.
We can apply this information to variable expressions. How can we do that?
Remember that a variable expression has variables, numbers, operations and sometimes exponents in it. We can start by thinking about evaluating variable expressions. When we evaluate a variable expression, we are finding the value of that expression. If given a value for a variable, we substitute the numerical value of the variable into the expression. Then we can find the total value of the expression.
Some variable expressions will have more than one variable in it. We can evaluate the expression if we have been given values for each of the variables.
Example
Evaluate the expression: \begin{align*}4b+12 \div 4-8\end{align*} if \begin{align*}b=6\end{align*}
First, we substitute the given value in for the variable.
\begin{align*}4(6)+ 12 \div 4-8\end{align*}
Next, we perform multiplication and division in order from left to right.
\begin{align*}24 + 3 - 8\end{align*}
Now we add and subtract in order from left to right.
\begin{align*}27 - 8 = 19\end{align*}
Our answer is 19.
That example had one variable in it. Here is an example that has two variables in it.
Example
Evaluate the expression: \begin{align*}\frac{21}{x}-7+8y\end{align*} if \begin{align*}x=3\end{align*} and \begin{align*}y=4\end{align*}
First, we substitute the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} into the expression.
\begin{align*}\frac{21}{3}-7+8(4)\end{align*}
Next, we perform multiplication and division in order from left to right. Notice that this problem has a fraction bar in it which means division.
\begin{align*}7-7+32\end{align*}
Now we add and subtract in order from left to right.
\begin{align*}0 + 32 = 32\end{align*}
Our answer is 32.
1I. Lesson Exercises
Evaluate the following variable expressions by following the order of operations.
- \begin{align*}5a+6 \div 2+9\end{align*} if \begin{align*}a\end{align*} is 5
- \begin{align*}7x+14 \div 7-3\end{align*} if \begin{align*}x\end{align*} is 4
- \begin{align*}\frac{48}{x}+ 5y-7\end{align*} if \begin{align*}x\end{align*} is 6 and \begin{align*}y\end{align*} is 9
Now take a few minutes to check your work with a peer.
II. Evaluate Variable Expressions Involving Powers and Grouping Symbols
Now that we know the basic order of operations, we can expand our rules to include evaluation of more complicated expressions. In intricate expressions, parentheses are used as grouping symbols. Parentheses indicate which operations should be done first. In the order of operations, operations within the parentheses are always first.
Remember how to simplify and evaluate exponents? \begin{align*}x^3 = xxx\end{align*}. Exponential notation is another factor we must account for in the order of operations. After completing the operations in parentheses, we then evaluate the exponents. Then we complete the multiplication and division from left to right; finally, we complete the addition and subtraction from left to right. The chart below shows the complete order of operations. If you keep the order of operations in the forefront of your mind and are careful to take each step one at a time and show your work, evaluating complex expressions using the order of operations will become second nature.
Let’s look at the order of operation once again.
Example
Evaluate the expression: \begin{align*}2x^2-(x+7)\end{align*} if \begin{align*}x=4\end{align*}
First, we substitute 4 in for \begin{align*}x\end{align*}.
\begin{align*}2(4^2)-(4+7)\end{align*}
Next, we add the terms in parentheses. Since parentheses is the first order of operations.
\begin{align*}2(4^2)-(11)\end{align*}
Now we can evaluate the exponent.
\begin{align*}2(16) - 11\end{align*}
Multiplication is next and there is no division.
\begin{align*}32 - 11\end{align*}
Finally, we complete with subtraction since there is no addition.
21
Our answer is 21.
Sometimes you will also evaluate expressions with more than one variable. Just keep track and follow the order of operations and you will be set.
1J. Lesson Exercises
- Evaluate \begin{align*}3x^2-2+(x+3)\end{align*} if \begin{align*}x\end{align*} is 2
- Evaluate \begin{align*}\frac{24}{x}+ (9-x)+y^2\end{align*} if \begin{align*}x\end{align*} is 3 and \begin{align*}y\end{align*} is 4
Take a few minutes to check your answers with a peer. Be sure to correct any errors that you might have made.
Be sure to write down the order of operations before moving on.
III. Write Variable Expressions to Represent and Solve Real-World Problems Using Order of Operations
When writing variable expressions to solve real-world problems, the most important part is translating the wording of the problem into a numeric expression. The first step is to read the problem in its entirety. Then we need to analyze the problem and decipher the mathematical relationships involved.
Example
Lydia and Bart both work at a bookstore. Lydia makes \begin{align*}x\end{align*} amount each hour; Bart has more experience, so he makes 1.5 times more than Lydia makes each hour. If Lydia and Bart both work 4 hours a day, how much will they make together in seven days if Lydia makes $8 an hour. Write an expression; then solve.
In this problem, \begin{align*}x\end{align*} stands for the amount Lydia makes per hour. Because Bart makes 1.5 times the amount that Lydia makes per hour, then \begin{align*}1.5x\end{align*} describes how much Bart makes per hour. The total Lydia and Bart make in 1 hour is therefore \begin{align*}(x + 1.5x)\end{align*}. Now the problem tells us that Lydia and Bart each work 4 hours per day. So, the amount they both make in 1 day is \begin{align*}4(x + 1.5x)\end{align*}. We want to find out how much they make in 7 days, so our expression is \begin{align*}7 \times 4(x + 1.5x)\end{align*}. The problem gives us the value of \begin{align*}x\end{align*}. Lydia makes $8 an hour. We substitute 8 for the variable in our expression and solve using the order of operations. Remember: parenthesis, exponents, multiplication, division, addition, subtraction.
\begin{align*}7 \times 4(x + 1.5x) &= 7 \times 4(8 + 1.5 \times 8)\\ 7 \times 4(x + 1.5x) &= 7 \times 4(8 + 12)\\ 7 \times 4(x + 1.5x) &= 7 \times 4(20)\\ 7 \times 4(x + 1.5x) &= 560\end{align*}
The answer is $560.00.
Remember to add the dollar sign, because the answer is in dollars.
Now let’s look at the introductory problem with our hikers and use what we have learned to solve this problem.
Real Life Example Completed
Dividing up the Groups
Here is the original problem once again. Reread it and underline any important information.
The first few days of Teen Adventure were spent at the base of the White Mountains in the Lafayette Place Campground. There the teens worked on survival skills, map skills, and basic first aid. The students were kept all together and Kelly made some terrific new friends. Her favorite part was all of the team building and trust building exercises that they learned. She even surprised herself by being able to complete some tasks that she wasn’t sure that she could do. Kelly and the other students learned to rely on themselves and each other over those first few days.
The night before their first hike, the group leaders told them that they would be divided up into hiking groups. Each week they would meet altogether once again to talk about their adventures, but during the actual week each group would be on a different trail. There would be three groups out of the 30 students. The 6 leaders would be divided into the 3 groups. For the first few days, there would be a manager along and two first-aid persons. After the first few days, the manager and the two first aid-persons would head back to the base office to be checked in with weekly or in case of emergency. So they would start with one number of people and after the first few days the number would change.
Kelly is curious about how many people will start and how many will be together after the first few days.
First, let’s work with the numbers involved.
30 students
6 group leaders
1 manager
2 first aid persons
Now we can write an expression to show how one group is created out of the whole.
\begin{align*}30 \div 3+6 \div 3\end{align*}
Next, we solve it for the number of students and leaders per group. Remember to follow the order of operations.
\begin{align*}10 + 2 = 12\end{align*} persons
For the first few days, there will also be a manager and two first aid persons.
\begin{align*}12 + 1 + 2\end{align*}
15 people in a group for the first few days.
Then the manager and two first aid persons leave.
\begin{align*}15 - 3 = 12\end{align*}
The core group will consist of 12 people.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Numerical Expression
- an expression that uses numbers and operations.
- Variable Expression
- an expression that uses numbers, variables and operations.
- Parentheses
- grouping symbols, the first step of the order of operations.
- Exponent
- the little number that tells how many times to multiply the base times itself.
- Order of Operations
- the order in which you perform each operation when evaluating an expression.
Technology Integration
Khan Academy, Order of Operations
James Sousa, Order of Operations - The Basics
Other Videos:
http://www.mathplayground.com/howto_pemdas.html – This is a video on order of operations known as PEMDAS.
Time to Practice
Directions: Use the order of operations to evaluate each numerical expression.
1. \begin{align*}5 + 3 \times 4\end{align*}
2. \begin{align*}6 \times 2 + 5 \times 3\end{align*}
3. \begin{align*}4 + 5 \times 2 - 9\end{align*}
4. \begin{align*}4 + 6 \div 2 + 10 - 3\end{align*}
5. \begin{align*}8 - 15 \div 3 + 4 \times 5\end{align*}
Directions: Evaluate the following variable expressions if \begin{align*}x=4, y=2, z=3\end{align*}
6. \begin{align*}x^2+ y\end{align*}
7. \begin{align*}2y^2+ 5-2\end{align*}
8. \begin{align*}x^2- y^2+ z\end{align*}
9. \begin{align*}3x^2+ 2x^2\end{align*}
10. \begin{align*}8+ x^2- 4y\end{align*}
11. \begin{align*}14 \div 2+ z^2- y\end{align*}
12. \begin{align*}20 + z^2-y\end{align*}
13. \begin{align*}5x-2y+3z\end{align*}
14. \begin{align*}5+(x-z)+ 5(6)\end{align*}
15. \begin{align*}8 + x-y^2+z\end{align*}
16. \begin{align*}(x+y)+ 5 \cdot 2 - 3\end{align*}
17. \begin{align*}4x^2+3z^3+ 2\end{align*}
Directions: Use what you have learned to answer the following questions as either true or false.
18. Parentheses are a grouping symbol.
19. Exponents can’t be evaluated unless the exponent is equal to 3.
20. If there is multiplication and division in a problem you always do the division first.