1.4: Writing Expressions
Introduction
Cookies at Lunch
“Here is a change that is great,” Cameron the eighth grade President said to the Student Council on Wednesday afternoon. “We can fundraise during lunch by selling cookies and we don’t need permission.”
“Really, that is a change,” Marcy commented.
The students at the middle school had always been able to fundraise, but they needed to get special permission. This was especially true regarding food and regarding lunch time. This news was greeted by some excitement and a lot of talking at the meeting.
“Hold on everybody,” Cameron shouted over the crowd. “Let’s talk about this.”
The group quieted down and Cameron began.
“We need $350.00 for the first autumn dance. We have $50.00 in our account right now. I figure that if we can see cookies for 25 cents each, then we can make about $30.00 per week in the two lunches,” Cameron proposed.
“I think we’ll make double that,” Tracy said smiling.
“Well maybe, but if we make the $30.00 then we will have the money for the autumn dance in no time. Just do the math!”
Did you do the math? Well, this lesson is about writing expressions. How long will it take based on Cameron’s plan? How long if Tracy is correct?
Pay attention and learn how to write expressions in this lesson. Then you will have a chance to put what you have learned to the test and “do the math” yourself.
What You Will Learn
By the end of this lesson you will be able to complete the following skills.
- Evaluate variable expressions with given values for the variables.
- Translate verbal phrases into variable expressions.
- Write and evaluate variable expressions for given problem situations.
- Write variable expressions to represent and solve real-world problems.
Teaching Time
I. Evaluate Variable Expressions with Given Values for the Variables
In the last few lessons you have been learning about evaluating expressions. This lesson is going to focus on writing and evaluating variable expressions. Think about the definition of a variable expression.
A variable expression is a group of numbers, operations and variables without using an equal sign. A variable is a letter used to represent an unknown quantity. A constant is a number without a variable.
Example
\begin{align*}6a+7\end{align*}
In this variable expression, \begin{align*}a\end{align*} is the variable and 7 is the constant.
Let’s look at evaluating variable expressions when we have been given a value for the variable.
Example
Evaluate the expression \begin{align*}4g + 1.5\end{align*} for \begin{align*}g = 8\end{align*}.
Step 1: Substitute 8 for the variable “\begin{align*}g\end{align*}.”
\begin{align*}4(8) + 1.5\end{align*}
Step 2: Follow the standard order of operations to solve: parentheses, exponents, multiply, divide, add, and then subtract.
\begin{align*}& 4(8) + 1.5 \ (\text{Multiply})\\ & 32 + 1.5 \ (\text{Add})\\ & 33.5\end{align*}
Our answer is 33.5
Example
Evaluate the expression \begin{align*}5ab + 2a - 7\end{align*}, when \begin{align*}a = 2\end{align*} and \begin{align*}b = 4\end{align*}.
Yes, there are. But don’t let that take you off course. If you simply substitute the given values into the expression and use the order of operations, you will end up with the correct answer.
Step 1: Substitute 2 for the variable “\begin{align*}a\end{align*}” and 4 for the variable “\begin{align*}b\end{align*}.”
\begin{align*}& 5ab + 2a - 7\\ & 5(2)(4) + 2(2) - 7\end{align*}
Step 2: Follow the standard order of operations to solve: parentheses, exponents, multiply, divide, add, and then subtract.
\begin{align*}& 5(2)(4) + 2(2) - 7 \ (\text{Multiply})\\ & 10(4) + 4 - 7\\ & 40 + 4 - 7 \ (\text{Add})\\ & 44 - 7 \ (\text{Subtract})\\ & 37\end{align*}
The answer is 37.
You have been working on evaluating expression for a couple of lessons. Now let’s look at writing variable expressions.
II. Translate Verbal Phrases into Variable Expressions
In this section, you will learn how to take a verbal phrase and write it as a variable expression. To accomplish this task, we need to think about what different words mean. A verbal expression is a mathematical statement that is expressed in words.
You will have to work as a detective to figure out what different words mean. Once you know what those words mean, you will be able to write different variable expressions.
Let’s start by looking at some mathematical operations written as words.
Addition
Sum
Plus
Increased by
More
Subtraction
Difference
Less than
Take away
Multiplication
Product
Times
Division
Quotient
Split up
This list does not include ALL of the ways to write the operations, but it will give you a good place to start.
Take a few minutes and write these words down in your notebook.
Now we can look at the following chart which starts with a verbal phrase and writes it as a variable expression.
Verbal Phrase | Variable Expression |
---|---|
Three minus a number | \begin{align*}3 - x\end{align*} |
A number increased by seven | \begin{align*}n + 7\end{align*} |
The difference between an unknown quantity and twenty-six | \begin{align*}s - 26\end{align*} |
A number decreased by nine | \begin{align*}w - 9\end{align*} |
Ten times a number plus four | \begin{align*}10f + 4\end{align*} |
Notice that words like “a number” and “an unknown quantity” let us know that we need to use a variable.
Now let’s look at working through a few of our own.
Example
Write a variable expression that reads “The product of a number and six plus four.”
A product is the answer in a multiplication problem. Therefore, this expression includes the operations multiplication and addition.
The answer is \begin{align*}6x+4\end{align*}
Example
Write a variable expression that reads “Ninety divided by a number minus eight.”
We could do this in several different ways. We could use a symbol, \begin{align*}\div\end{align*}, to show division or we could use a fraction bar to show division. Because you are moving toward Algebra, let’s use a fraction bar.
The answer is \begin{align*}\frac{90}{b}-8\end{align*}.
We can also write variable expressions that include grouping symbols. This can be a little trickier, but if you look for clues, you can figure these verbal expressions out too.
Example
Write a variable expression that reads “Two less than a number, multiplied by thirty-six.”
The clue word less tells you to subtract the number and two. Since this is a two-step problem, you can place grouping symbols around the first part of the expression.
The answer is \begin{align*}36(n - 2)\end{align*}.
You can figure out how to write a verbal expression as a variable expression by decoding the words in the verbal expression into their mathematical meaning. Now let’s look at writing variable expressions for given situations.
III. Write and Evaluate Variable Expressions for Given Problem Situations
Now that you have learned how to write variable expressions, you can apply this to evaluating given expressions. To complete this section, you will need to write a variable expression from a verbal expression and then evaluate the expression using the given value.
Example
Evaluate four times a number minus five, when the number is four.
First, we need to write an expression. We know that there are two operations in this expression. The word “times” tells us that we have multiplication and the word “minus” tells us that we have subtraction. We also know that the value of the variable is four.
\begin{align*}4n-5, \ when \ n \ is \ 4\end{align*}
Next, we can substitute the given value in for \begin{align*}n\end{align*} and evaluate the expression.
\begin{align*}& 4(4)-5\\ & 16-5\\ & 11\end{align*}
The answer is 11.
Example
Evaluate a number squared plus six when the number is eight.
First, notice that we have the word “squared” that lets us know that we will be working with a power. Then we have the word “plus” so we know that our second operation is addition. The unknown value is eight.
\begin{align*}x^2+6 \ when \ x \ is \ 8\end{align*}
Next, we substitute the 8 in for the value of the variable.
\begin{align*}& 8^2+6\\ & 64 + 6\\ & 70\end{align*}
The answer is 70.
Sometimes a variable expression can be used to represent a real-world situation. Let’s look at how to write and evaluate variable expressions that are connected with real-world scenarios.
IV. Write Variable Expressions to Represent and Solve Real-World Problems
A variable expression can be very helpful when problem solving. We can work to use a variable expression to understand a given situation. The variable expression can also help us to work through something and find a solution.
Example
Grace is saving money to purchase a new bike that costs one hundred seventy-five dollars. Grace has twenty-five dollars and is also saving twenty dollars each week. Write and solve a variable expression to determine the number of weeks “\begin{align*}w\end{align*}” it will take Grace to save for her new bike.
First, add the amount that Grace has already saved, $25.00, to the amount that she plans on saving each week. Because the number of weeks is unknown, we multiply the amount she is planning on saving each week, $20.00 by the variable “\begin{align*}w\end{align*}”. The “\begin{align*}w\end{align*}” represents the number of weeks. Set the expression equal to Grace’s goal of $175.00.
\begin{align*}25.00 + 20.00w = 175.00\end{align*}
Now we know that she needs to earn an amount of money that equals $175.00. Because she has already earned 25.00, we can subtract that from the total she needs.
\begin{align*}175.00 - 25.00 = \$150.00\end{align*}
We are left with the expression.
\begin{align*}20.00w = 150.00\end{align*}
Solve for “\begin{align*}w\end{align*}” by completing the inverse of multiplication. Since the number of weeks “\begin{align*}w\end{align*}” times 20.00 equals 150.00, divide 150.00 by \begin{align*}20.00w\end{align*}.
\begin{align*}\frac{20.00w}{20.00}=\frac{150.00}{20.00}=7.5\end{align*}
Therefore, it will take Grace seven and one half weeks to save for her bike.
You can find problems like this one all the time in real-life situations. Now let’s go back to the problem in the introduction and use what we have learned to solve the problem.
Real-Life Example Completed
Cookies at Lunch
Here is the original problem once again. Reread it and then write an expression to show Cameron’s proposal. How many weeks will it take based on Cameron’s plan? If Tracy is correct, how many weeks will it take then? There are three parts to your answer.
“Here is a change that is great,” Cameron the eighth grade President said to the Student Council on Wednesday afternoon. “We can fundraise during lunch by selling cookies and we don’t need permission.”
“Really, that is a change,” Marcy commented.
The students at the middle school had always been able to fundraise, but they needed to get special permission. This was especially true regarding food and regarding lunch time. This news was greeted by some excitement and a lot of talking at the meeting.
“Hold on everybody,” Cameron shouted over the crowd. “Let’s talk about this.”
The group quieted down and Cameron began.
“We need $350.00 for the first autumn dance. We have $50.00 in our account right now. I figure that if we can see cookies for 25 cents each, then we can make about $30.00 per week in the two lunches,” Cameron proposed.
“I think we’ll make double that,” Tracy said smiling.
“Well maybe, but if we make the $30.00 then we will have the money for the autumn dance in no time. Just do the math!”
Take what you have learned and write an expression. Then evaluate the expression for the two answers. Remember that there are three parts to your answer.
Solution to Real – Life Example
To work on this problem, let’s first think about the given information and write an expression. We use Cameron’s information to do this.
There is $50.00 in the student council account.
Cameron thinks that they can save $30.00 per week.
The number of weeks needed to save the money is unknown that is our variable, \begin{align*}w\end{align*}.
Here is the expression.
\begin{align*}50 + 30w\end{align*}
Next, we can write an equation using this expression to solve for the number of weeks needed to make $350.00.
\begin{align*}50 + 30w = 350\end{align*}
We can solve it by using mental math.
\begin{align*}w = 10\end{align*}
It will take 10 weeks based on Cameron’s proposal.
If Tracy is correct and the students are able to make twice as much money, then it will take them 5 weeks to make the money.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Variable Expression
- a group of numbers, operations and variables without an equal sign.
- Variable
- a letter used to represent an unknown number
- Constant
- a number in an expression that does not have a variable.
- Verbal Expression
- using language to write a mathematical expression instead of numbers, symbols and variables.
Time to Practice
Directions: Evaluate each variable expression using the given values for each variable.
- \begin{align*}6a+7\end{align*} when \begin{align*}a\end{align*} is 8
- \begin{align*}9x-y\end{align*} when \begin{align*}x\end{align*} is 2 and \begin{align*}y\end{align*} is 4.
- \begin{align*}5a+ b^2\end{align*} when \begin{align*}a\end{align*} is 12 and \begin{align*}b\end{align*} is 4.
- \begin{align*}\frac{8}{x}+2\end{align*} when \begin{align*}x\end{align*} is 4
- \begin{align*}6x+2.5\end{align*} when \begin{align*}x\end{align*} is 2.
- \begin{align*}y^2+4\end{align*} when \begin{align*}y\end{align*} is 9
- \begin{align*}7x+2y\end{align*} when \begin{align*}x\end{align*} is 3 and \begin{align*}y\end{align*} is 5
- \begin{align*}9xy+ x^2\end{align*} when \begin{align*}x\end{align*} is 4 and \begin{align*}y\end{align*} is 2
- \begin{align*}3ab+ b^3\end{align*} when \begin{align*}a\end{align*} is 9 and \begin{align*}b\end{align*} is 2
- \begin{align*}16xy^2+ 14\end{align*} when \begin{align*}x\end{align*} is 3 and \begin{align*}y\end{align*} is 4
Directions: Write a variable expression for each verbal expression.
- The sum of a number and twelve.
- The difference between a number and eight.
- Three times a number
- A number squared plus five
- A number divided by two plus seven
- Four times the quantity of a number plus six
- A number times two divided by four
- A number times six plus the same number times two
- A number squared plus seven take a way four
- A number divided by three plus twelve
Directions: For numbers evaluate each expressions.
- The sum of a number 12 and twelve.
- The difference between a number 12 and eight.
- Three times a number 12
- A number 12 squared plus five
- A number 12 divided by two plus seven
- Four times the quantity of a number 12 plus six
- A number 12 times two divided by four
- A number 12 times six plus the same number times two
- A number 12 squared plus seven take a way four
- A number 12 divided by three plus twelve