Have you ever had a summer job? Were you able to save a lot of money?
Joshua has gotten a job at the city zoo for the summer. His love of animals prompted him to seek out the job, and he was so excited when the zoo called to say that he had been hired for the job. There is only one problem, in his excitement, Joshua forgot to ask how much he would be paid per hour.
Joshua thinks he remembers that the manager said he would be paid either $8.00 an hour or $9.00 an hour. Joshua is planning to work 20 hours per week all summer.
He writes the following expression on a piece of paper.
\begin{align*}20x = ?\end{align*}
Now he isn't sure what to do. He knows that the unknown is the amount of money.
Can you help Joshua figure out how much he might earn?
Use what you learn in this Concept to figure out how much Joshua will earn if he is paid eight dollars an hour, or how much he will earn if he is paid nine dollars an hour.
Guidance
In this Concept we begin with a new definition that we haven’t talked about before. It is the definition of a variable.
What is a variable?
A variable is a letter that is used to represent an unknown quantity.
Often we use \begin{align*}x\end{align*} or \begin{align*}y\end{align*} to represent the unknown quantity, but any letter can be used as a variable.
Here are some variables.
\begin{align*}a\end{align*}
\begin{align*}b\end{align*}
\begin{align*}c\end{align*}
Notice that the variables here are all lowercase letters. This is often the case with variables.
A variable can be used in any sort of mathematical expression.
A variable expression is an expression with one or more operations that has variables but no equals sign.
This means that we can have expressions and variable expressions. When we have a variable expression, we have an expression with one or more operations and variables too.
To understand variable expressions a little better, let’s think about some ways that we can show addition, subtraction, multiplication and division in mathematics. Addition can be shown by using a \begin{align*}+\end{align*} sign. Subtraction can be shown using a subtraction or minus sign \begin{align*}-\end{align*} . Multiplication can be shown a couple of different ways.
- We can use a times symbol as in \begin{align*}5 \times 6 = 30.\end{align*}
- We can use two sets of parentheses. \begin{align*}(5)(6) = 30\end{align*}
- We can use a variable next to a number. \begin{align*}6x\end{align*} means 6 times the unknown \begin{align*}x\end{align*}.
- We can use one number next to parentheses. \begin{align*}4(3) = 12\end{align*}
Division can be shown in a couple of different ways.
- We can use the division sign. \begin{align*} \div \end{align*}
- We can use the fraction bar. \begin{align*}\frac{6}{2}\end{align*} means \begin{align*}6 \div 2\end{align*}
Now that you are in sixth grade, you will begin to see operations shown in different ways.
Let’s go back to variable expressions.
It is actually easy to evaluate different variable expressions when we have a given value for the variable.
Evaluate \begin{align*}5+a\end{align*}, when \begin{align*}a = 18\end{align*}.
Here we are going to substitute our given value for the variable. In this case, we substitute 18 in for \begin{align*}a\end{align*} and then add.
\begin{align*}& 5 + 18\\ & 23\end{align*}
Our answer is 23.
We can evaluate any variable expression as long as we have been given a value for the variable.
Evaluate \begin{align*}b-22\end{align*} when \begin{align*}b\end{align*} is 40. Next, we complete the subtraction by substituting our given value 40 into the expression for \begin{align*}b\end{align*}.
\begin{align*}& 40 - 22 \\ & 18\end{align*}
Our answer is 18.
Evaluate \begin{align*}7x\end{align*} when \begin{align*}x\end{align*} is 12. This is a multiplication problem. We substitute our given value in for \begin{align*}x\end{align*} and then multiply.
\begin{align*}& 7(12)\\ & 84\end{align*}
Now it is time for you to try a few on your own. Evaluate each expression using the given value.
Example A
Evaluate \begin{align*}17+y\end{align*} when \begin{align*}y\end{align*} is 12.
Solution: 29
Example B
Evaluate \begin{align*}5c\end{align*} when \begin{align*}c\end{align*} is 9.
Solution: 45
Example C
Evaluate \begin{align*}8 \div x\end{align*} when \begin{align*}x\end{align*} is 4.
Solution: 2
Now we can help Joshua. Here is the original problem once again.
Joshua has gotten a job at the city zoo for the summer. His love of animals prompted him to seek out the job, and he was so excited when the zoo called to say that he had been hired for the job. There is only one problem, in his excitement, Joshua forgot to ask how much he would be paid per hour.
Joshua thinks he remembers that the manager said he would be paid either $8.00 an hour or $9.00 an hour. Joshua is planning to work 20 hours per week all summer.
He writes the following expression on a piece of paper.
\begin{align*}20x = ?\end{align*}
Our given values for the unknown, x, are eight dollars and nine dollars. Let's work with eight dollars first. We can substitute 8 into the equation for x.
\begin{align*}20 \times 8 = ?\end{align*}
Next we multiply.
\begin{align*}160\end{align*}
At eight dollars an hour, Joshua will earn $160.00 per week.
Now let's try nine.
\begin{align*}20 \times 9 = ?\end{align*}
\begin{align*}180\end{align*}
At nine dollars an hour, Joshua will earn $180.00 per week.
Vocabulary
- Evaluate
- to simplify an expression that does not have an equals sign.
- Variable
- a letter, usually lowercase, that is used to represent an unknown quantity.
- Expression
- a number sentence that uses operations but does not have an equals sign
- Variable Expression
- a number sentence that has variables or unknown quantities in it with one or more operations and no equals sign.
Guided Practice
Here is one for you to try on your own.
Evaluate \begin{align*}\frac{14}{x}\end{align*} when \begin{align*}x\end{align*} is 2.
Answer
Here we have a fraction bar which tells us that this is a division problem. We substitute the given value in for \begin{align*}x\end{align*} and divide.
\begin{align*}\frac{14}{2} = 7\end{align*}
This is our answer.
Video Review
Khan Academy Evaluating an Expression
James Sousa Example of Evaluating an Expression
James Sousa Example of Evaluating an Expression
Practice
Directions: Evaluate each of the variable expressions when \begin{align*}a = 4, \ b = 5, \ c = 6\end{align*}
1. \begin{align*}5 + a\end{align*}
2. \begin{align*}6 + b\end{align*}
3. \begin{align*}7 + c\end{align*}
4. \begin{align*}8 - a\end{align*}
5. \begin{align*}9c\end{align*}
6. \begin{align*}10a\end{align*}
7. \begin{align*}7c\end{align*}
8. \begin{align*}9a\end{align*}
9. \begin{align*}4b\end{align*}
10. \begin{align*}\frac{16}{a}\end{align*}
11. \begin{align*}\frac{42}{c}\end{align*}
12. \begin{align*}\frac{c}{2}\end{align*}
13. \begin{align*}15a\end{align*}
14. \begin{align*}9b\end{align*}
15. \begin{align*}\frac{15}{b}\end{align*}