1.6: Patterns and Expressions
Jeremy read that degrees Celsius converted to degrees Fahrenheit is "the sum of 32 and \begin{align*}\frac{9}{5}\end{align*}
Guidance
In mathematics, especially in algebra, we look for patterns in the numbers that we see. Using mathematical verbs and variables, expressions can be written to describe a pattern. An algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations. We can describe patterns using phrases as well, and we want to be able to translate these phrases into algebraic expressions.
Consider a theme park charging an admission of $28 per person. A rule can be written to describe the relationship between the amount of money taken at the ticket booth and the number of people entering the park. In words, the relationship can be stated as “The money taken in dollars is (equals) twentyeight times the number of people who enter the park.”
The English phrase above can be translated (written in another language) into an algebraic expression. Using mathematical verbs and nouns learned from previous lessons, any phrase can be written as an algebraic expression.
Example A
Write an algebraic expression for the following phrase.
The product of \begin{align*}c\end{align*}
Solution:
The verb is product, meaning “to multiply.” Therefore, the phrase is asking for the answer found by multiplying \begin{align*}c\end{align*}
Example B
Write an expression to describe the amount of revenue of the theme park described above.
Solution:
An appropriate variable to describe the number of people could be \begin{align*}p\end{align*}
Some phrases are harder to translate than others.
Example C
Translate the phrase "5 less than 2 times a number."
Solution:
The word less lets you know that you are going to take away, or subtract, a number. Many students will want to turn this expression into \begin{align*}52n\end{align*}
Guided Practice
A student organization sells shirts to raise money for events and activities. The shirts are printed with the organization's logo and the total costs are $100 plus $7 for each shirt. The students sell the shirts for $15 each. Write an expression for the cost and an expression for the revenue (money earned).
Solution:
We can use \begin{align*}x\end{align*}
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Patterns and Equations (13:18)
For exercises 1 – 15, translate the English phrase into an algebraic expression. For the exercises without a stated variable, choose a letter to represent the unknown quantity.
 Sixteen more than a number
 The quotient of \begin{align*}h\end{align*}
h and 8  Fortytwo less than \begin{align*}y\end{align*}
y  The product of \begin{align*}k\end{align*}
k and three  The sum of \begin{align*}g\end{align*}
g and \begin{align*}7\end{align*}−7 
\begin{align*}r\end{align*}
r minus 5.8  6 more than 5 times a number
 6 divided by a number minus 12
 A number divided by \begin{align*}11\end{align*}
−11  27 less than a number times four
 The quotient of 9.6 and \begin{align*}m\end{align*}
m  2 less than 10 times a number
 The quotient of \begin{align*}d\end{align*}
d and five times \begin{align*}s\end{align*}s  35 less than \begin{align*}x\end{align*}
x  The product of 6, \begin{align*}9\end{align*}
−9 , and \begin{align*}u\end{align*}u
In exercises 16 – 24, write an English phrase for each algebraic expression

\begin{align*}J  9\end{align*}
J−9 
\begin{align*}\frac{n}{14}\end{align*}
n14 
\begin{align*}17a\end{align*}
17−a 
\begin{align*}3l16\end{align*}
3l−16 
\begin{align*}\frac{1}{2} (h)(b)\end{align*}
12(h)(b) 
\begin{align*}\frac{b}{3} + \frac{z}{2}\end{align*}
b3+z2 
\begin{align*}4.72f\end{align*}
4.7−2f 
\begin{align*}5.8 + k\end{align*}
5.8+k 
\begin{align*}2l+2w\end{align*}
2l+2w
In exercises 25 – 28, define a variable to represent the unknown quantity and write an expression to describe the situation.
 The unit cost represents the quotient of the total cost and number of items purchased. Write an expression to represent the unit cost of the following: The total cost is $14.50 for \begin{align*}n\end{align*}
n objects.  The area of a square is the side length squared.
 The total length of ribbon needed to make dance outfits is 15 times the number of outfits.
 What is the remaining amount of chocolate squares if you started with 16 and have eaten some?
Use your sense of variables and operations to answer the following questions.
 Describe a realworld situation that can be represented by \begin{align*}h + 9\end{align*}
h+9 .  What is the difference between \begin{align*}\frac{7}{m}\end{align*}
7m and \begin{align*}\frac{m}{7}\end{align*}m7 ?
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Term  Definition 

algebraic expression  An algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations. 
The symbol means "therefore" or "because of this."  
Algebraic Equation  An algebraic equation contains numbers, variables, operations, and an equals sign. 
domain  The domain of a function is the set of values for which the function is defined. 
Equation  An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs. 
horizontal axis  The horizontal axis is also referred to as the axis of a coordinate graph. By convention, we graph the input variable on the axis. 
Range  The range of a function is the set of values for which the function is defined. 
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. 
vertical axis  The vertical axis is also referred to as the axis of a coordinate graph. By convention, we graph the output variable on the axis. 
Image Attributions
Here you'll learn how to take an English phrase and produce an equivalent algebraic expression. You'll also practice taking an algebraic expression and producing an equivalent English phrase.