1.5: Variables and Expressions
Introduction
The Ticket Revenue Dilemma
Like many of his friends, Joshua has a summer job at the city zoo. Joshua loves people and so he is working at the ticket counter. His job is to count the people entering the zoo each day. He does this twice. He counts them in the morning and in the afternoon. Sometimes he has more people come in the morning and sometimes the counts are higher in the afternoon.
Joshua loves his job. He loves figuring out how much money the zoo has made from the ticket sales. Joshua has a thing for mental math. While many of his friends think it is too difficult, Joshua enjoys figuring it out in his head.
To enter the zoo for the day, it costs an adult \begin{align*}\$7.00\end{align*} and a child \begin{align*}\$5.00\end{align*}.
Joshua has written the following expression to help him to figure out the amount of money that the zoo makes in half a day. He divides his arithmetic up between the morning and the afternoon.
\begin{align*}7x+5y\end{align*}
Here are his counts for Monday.
AM - 65 adults and 75 children
PM - 35 adults and 50 children
Here are his counts for Tuesday.
AM - 70 adults and 85 children
PM - 50 adults and 35 children
Given these counts, how much revenue (money) was collected at the zoo for the entire day on Monday?
How much money was collected at the zoo for the entire day on Tuesday?
How much money was collected in the two days combined?
Joshua can figure this out using his expression.
Can you? In this lesson, you will learn how to use a variable expression to solve a real-world problem.
Pay close attention. You will need these skills to figure out the zoo revenue for Monday and Tuesday.
What You Will Learn
In this lesson, you will learn to use the following skills:
- Evaluating single variable expressions with given values for the variable
- Evaluating multi-variable expressions with given values for the variable
- Using given expressions to analyze and solve real-world problems
Teaching Time
I. Evaluating Single Variable Expressions with Given Values for the Variable
In this lesson we begin with a new concept that we haven’t talked about before. It is the concept 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 examples of 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.
Here is an example.
Example
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.
Example
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.
Example
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*}
Example
Evaluate \begin{align*}\frac{14}{x}\end{align*} when \begin{align*}x\end{align*} is 2.
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*}
Here we have looked at several different examples that all had one variable and one operation.
Now it is time for you to try a few on your own. Evaluate each expression using the given value.
- Evaluate \begin{align*}17+y\end{align*} when \begin{align*}y\end{align*} is 12.
- Evaluate \begin{align*}5c\end{align*} when \begin{align*}c\end{align*} is 9.
- Evaluate \begin{align*}8 \div x\end{align*} when \begin{align*}x\end{align*} is 4.
Take a minute and check your work with a peer.
II. Evaluating Multi-Variable Expressions with Given Values
In the last section, we evaluated expressions that had one variable and one operation.
In this section, we are going to be working with expressions that have multiple variables and multiple operations.
Let’s look at an example to see what this looks like.
Example
Evaluate \begin{align*} 6a+b \end{align*} when \begin{align*}a\end{align*} is 4 and \begin{align*}b\end{align*} is 6. First, of all, you can see that there are two variables in this expression. There are also two operations here. The first one is multiplication: \begin{align*}6a\end{align*} lets us know that we are going to multiply 6 times the value of \begin{align*}a\end{align*}. The second one is addition: the \begin{align*}+ \ b\end{align*} lets us know that we are going to add the value of \begin{align*}b\end{align*}. We have also been given the values of \begin{align*}a\end{align*} and \begin{align*}b\end{align*}. We substitute the given values for each variable into the expression and evaluate it.
\begin{align*}& 6(4)+5\\ & 24+5\\ & 29\end{align*}
Our answer is 29.
Notice that we used the order of operations when working through this problem.
Order of Operations
P - parentheses
E - exponents
MD - multiplication and division in order from left to right
AS - addition and subtraction in order from left to right
Whenever we are evaluating expressions with more than one operation in them, always refer back and use the order of operations.
Let’s look at another example with multiple variables and expressions.
Example
Evaluate \begin{align*} 7b-d \end{align*} when \begin{align*}b\end{align*} is 7 and \begin{align*}d\end{align*} is 11.
First, we substitute the given values in for the variables.
\begin{align*}& 7(7) - 11\\ & 49 - 11\\ & 38\end{align*}
Our answer is 38.
What about when we have an example that is all variables?
Example
Evaluate \begin{align*} ab + cd \end{align*} when \begin{align*}a\end{align*} is 4, \begin{align*}b\end{align*} is 3, \begin{align*}c\end{align*} is 10 and \begin{align*}d\end{align*} is 6.
We work on this one in the same way as the other examples.
Begin by substituting the given values in for the variables.
\begin{align*}(4)(3) + (10)(6)\end{align*}
We have two multiplication problems here and one addition.
Next, we follow the order of operations to evaluate the expression.
\begin{align*}& 12 + 60\\ & 72\end{align*}
Our answer is 72.
Now it is time for you to try a few on your own.
- Evaluate \begin{align*}12x - y \end{align*} when \begin{align*}x\end{align*} is 4 and \begin{align*}y\end{align*} is 9.
- Evaluate \begin{align*}\frac{12}{a} + 4\end{align*} when \begin{align*}a\end{align*} is 3.
- Evaluate \begin{align*}5x + 3y \end{align*} when \begin{align*}x\end{align*} is 4 and \begin{align*}y\end{align*} is 8.
Take a minute to check your answers with a peer.
III. Using Given Expressions to Analyze and Solve Real-World Problems
Our dilemma at the beginning of this lesson is an excellent way to see how given expressions can be used to analyze real-world problems.
Before we complete the problem of the zoo revenue, let’s look at one other problem.
Our example works with money because our original problem works with money too.
Example
Joanne has a pile of nickels and a pile of dimes. She counts her money and figures out that she has 25 nickels and 36 dimes. Given these counts, how much money does Joanne have in all?
The first thing that we need to do is to underline all of the important information in the problem.
Joanne has a pile of nickels and a pile of dimes. She counts her money and figures out that she has 25 nickels and 36 dimes. Given these counts, how much money does Joanne have in all?
Next, we need to write an expression with a variable.
\begin{align*} .05x +.10y \end{align*}
A nickel is 5 cents. We can use decimal .05 to show that amount in dollars.
A dime is 10 cents. We can use decimal .10 to show that amount in dollars.
The \begin{align*}x\end{align*} represents the number of nickels.
The \begin{align*}y\end{align*} represents the number of dimes.
We have been given the number of dimes and nickels that Joanne has.
We can substitute those values into our expression for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
Example
\begin{align*}.05(25) + .10(36)\end{align*}
Next, we evaluate the expression.
\begin{align*}1.25 + 3.60 = \$4.50\end{align*}
Joanne has \begin{align*}\$4.50\end{align*} total. You can see why we changed the way we wrote the value of coins from cents to dollars now, because our answer is in dollars.
Now let’s go and complete our dilemma from the beginning of the lesson.
Real Life Example Completed
The Ticket Revenue Dilemma
Let’s use what we have learned about variable expressions and given values to solve the revenue question from the beginning of the lesson.
Here is the problem once again:
Like many of his friends, Joshua has a summer job at the city zoo. Joshua loves people and so he is working at the ticket counter. His job is to count the people entering the zoo each day. He does this twice. He counts them in the morning and in the afternoon. Sometimes he has more people come in the morning and sometimes the counts are higher in the afternoon.
Joshua loves his job. He loves figuring out how much money the zoo has made from the ticket sales. Joshua has a thing for mental math. While many of his friends think it is too difficult, Joshua enjoys figuring it out in his head.
To enter the zoo for the day, it costs an adult \begin{align*}\$7.00\end{align*} and a child \begin{align*}\$5.00\end{align*}.
Joshua has written the following expression to help him to figure out the amount of money that the zoo makes in half a day. He divides his arithmetic up between the morning and the afternoon.
\begin{align*}\underline{7x+5y}\end{align*}
Here are his counts for Monday.
AM - 65 adults and 75 children
PM - 35 adults and 50 children
Here are his counts for Tuesday.
AM - 70 adults and 85 children
PM - 50 adults and 35 children
Begin by underlining all of the important information in the problem, done here already.
First, we can start with Monday.
Our expression remains the same.
We can use \begin{align*} 7x+5y \end{align*}.
For Monday morning, the zoo had 65 adults and 75 children visit. Those are the given values that we can substitute into our expression for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
\begin{align*}& 7(65)+5(75)\\ & 455+375\\ & \$830.00\end{align*} For Monday afternoon, the zoo had 35 adults and 50 children visit. Those are the given values that we can substitute into our expression for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
\begin{align*}& 7(35)+5(50) \\ & 245+250\\ & \$495.00\end{align*}
The total amount of money made on Monday is \begin{align*}830 + 495 = \$1325.\end{align*}
Next, we can figure out Tuesday.
For Tuesday morning, the zoo had 70 adults and 85 children visit. Those are the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
\begin{align*}& 7(70)+5(85)\\ & 490+425\\ & \$915.00\end{align*}
For Tuesday afternoon, the zoo had 50 adults and 35 children visit. Those are the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
\begin{align*}& 7(50)+5(35)\\ & 350+175\\ & \$525\end{align*}
The total amount of money made on Tuesday is \begin{align*}915 + 525 = \$1440\end{align*}.
If we wanted to figure out the total amount of revenue for both days combined, we simply add the two totals together.
\begin{align*} \$1325 + \$1440 = \$2765.00\end{align*}
Vocabulary
Here are the vocabulary words that have been used in this lesson.
- 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.
- Revenue
- means money
Technology Integration
Here's a preview of evaluating expressions from a high-school algebra course.
Khan Academy Evaluating an Expression
James Sousa Example of Evaluating an Expression
James Sousa Example of Evaluating an Expression
This is a video with a jingle to help you remember variables and expressions. Very entertaining!
http://www.harcourtschool.com/jingles/jingles_all/35mystery_number.html -
- http://www.harcourtschool.com/jingles/jingles_all/35mystery_number.html - This is a video with a jingle to help you remember variables and expressions. Very entertaining!
Time to 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*}
Directions: Evaluate each multi-variable expression when \begin{align*}x = 2 \end{align*} and \begin{align*}y = 3\end{align*}.
16. \begin{align*}2x + y\end{align*}
17. \begin{align*}9x - y \end{align*}
18. \begin{align*}x + y \end{align*}
19. \begin{align*}xy \end{align*}
20. \begin{align*}xy + 3 \end{align*}
21. \begin{align*}9y - 5 \end{align*}
22. \begin{align*}10x - 2y \end{align*}
23. \begin{align*}3x + 6y \end{align*}
24. \begin{align*}2x + 2y \end{align*}
25. \begin{align*}7x - 3y \end{align*}
26. \begin{align*}3y - 2 \end{align*}
27. \begin{align*}10x - 8 \end{align*}
28. \begin{align*}12x - 3y \end{align*}
29. \begin{align*}9x + 7y \end{align*}
30. \begin{align*}11x - 7y \end{align*}
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