1.3: Patterns and Expressions
In mathematics, especially in algebra, we look for patterns in the numbers that we see. Using mathematical verbs and variables studied in lessons 1.1 and 1.2, expressions can be written to describe a pattern.
Definition: An algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations.
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) twenty-eight 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 sentence can be written as an algebraic expression.
Example 1: Write an algebraic expression for the following phrase.
The product of \begin{align*}c\end{align*} and 4.
Solution: The verb is product, meaning “to multiply.” Therefore, the phrase is asking for the answer found by multiplying \begin{align*}c\end{align*} and 4. The nouns are the number 4 and the variable \begin{align*}c\end{align*}. The expression becomes \begin{align*}4 \times c, \ 4(c)\end{align*}, or using shorthand, \begin{align*}4c\end{align*}.
Example 2: Write an expression to describe the amount of revenue of the theme park.
Solution: An appropriate variable to describe the number of people could be \begin{align*}p\end{align*}. Rewriting the English phrase into a mathematical phrase, it becomes \begin{align*}28 \times p\end{align*}.
Using Words to Describe Patterns
Sometimes patterns are given in tabular format (meaning presented in a table). An important job of analysts is to describe a pattern so others can understand it.
Example 3: Using the table below, describe the pattern in words.
\begin{align*}&&x && -1 && 0 && 1 && 2 && 3 && 4\\ &&y && -5 && 0 && 5 && 10 && 15 && 20\end{align*}
Solution: We can see from the table that \begin{align*}y\end{align*} is five times bigger than \begin{align*}x\end{align*}. Therefore, the pattern is that the “\begin{align*}y\end{align*} value is five times larger than the \begin{align*}x\end{align*} value.”
Example 4: Using the table below, describe the pattern in words and in an expression.
Zarina has a $100 gift card and has been spending money in small regular amounts. She checks the balance on the card weekly, and records the balance in the following table.
Week # | Balance ($) |
---|---|
1 | 100 |
2 | 78 |
3 | 56 |
4 | 34 |
Solution: Each week the amount of her gift card is $22 less than the week before. The pattern in words is: “The gift card started at $100 and is decreasing by $22 each week.”
The expression found in example 4 can be used to answer many situations. Suppose, for instance, that Zarina has been using her gift card for 4 weeks. By substituting the number 4 for the variable \begin{align*}w\end{align*}, it can be determined that Zarina has $12 left on her gift card.
Solution:
\begin{align*}100-22w\end{align*}
When \begin{align*}w = 4\end{align*}, the expression becomes
\begin{align*}&100-22(4)\\ &100-88\\ &12\end{align*}
After 4 weeks, Zarina has $12 left on her gift card.
Practice Set
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. CK-12 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*} and 8
- Forty-two less than \begin{align*}y\end{align*}
- The product of \begin{align*}k\end{align*} and three
- The sum of \begin{align*}g\end{align*} and \begin{align*}-7\end{align*}
- \begin{align*}r\end{align*} 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*}
- 27 less than a number times four
- The quotient of 9.6 and \begin{align*}m\end{align*}
- 2 less than 10 times a number
- The quotient of \begin{align*}d\end{align*} and five times \begin{align*}s\end{align*}
- 35 less than \begin{align*}x\end{align*}
- The product of 6, \begin{align*}-9\end{align*}, and \begin{align*}u\end{align*}
In exercises 16 – 24, write an English phrase for each algebraic expression
- \begin{align*}J - 9\end{align*}
- \begin{align*}\frac{n}{14}\end{align*}
- \begin{align*}17-a\end{align*}
- \begin{align*}3l-16\end{align*}
- \begin{align*}\frac{1}{2} (h)(b)\end{align*}
- \begin{align*}\frac{b}{3} + \frac{z}{2}\end{align*}
- \begin{align*}4.7-2f\end{align*}
- \begin{align*}5.8 + k\end{align*}
- \begin{align*}2l+2w\end{align*}
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*} 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?
- Describe a real-world situation that can be represented by \begin{align*}h + 9\end{align*}.
- What is the difference between \begin{align*}\frac{7}{m}\end{align*} and \begin{align*}\frac{m}{7}\end{align*}?
In questions 31 – 34, write the pattern of the table: a) in words and b) with an algebraic expression.
- Number of workers and number of video games packaged
\begin{align*}&\text{People} && 0 && 1 && 2 && 5 && 10 && 50 && 200\\ &\text{Amount} && 0 && 65 && 87 && 109 && 131 && 153 && 175\end{align*}
- The number of hours worked and the total pay
\begin{align*}&\text{Hours} && 1 && 2 && 3 && 4 && 5 && 6\\ &\text{Total Pay} && 15 && 22 && 29 && 36 && 43 && 50\end{align*}
- The number of hours of an experiment and the total number of bacteria
\begin{align*}&\text{Hours} && 0 && 1 && 2 && 5 && 10\\ &\text{Bacteria} && 0 && 2 && 4 && 32 && 1024\end{align*}
- With each filled seat, the number of people on a Ferris wheel doubles.
- Write an expression to describe this situation.
- How many people are on a Ferris wheel with 17 seats filled?
- Using the theme park situation from the lesson, how much revenue would be generated by 2,518 people?
Mixed Review
- Use parentheses to make the equation true: \begin{align*}10+6 \div 2-3=5\end{align*}.
- Find the value of \begin{align*}5x^2 - 4y\end{align*} for \begin{align*}x = -4\end{align*} and \begin{align*}y = 5\end{align*}.
- Find the value of \begin{align*}\frac{x^2y^3}{x^3 + y^2}\end{align*} for \begin{align*}x = 2\end{align*} and \begin{align*}y=-4\end{align*}.
- Simplify: \begin{align*}2 - (t - 7)^2 \times (u^3 - v)\end{align*} when \begin{align*}t = 19, u = 4\end{align*}, and \begin{align*}v = 2\end{align*}.
- Simplify: \begin{align*}2 - (19 - 7)^2 \times (4^3 - 2)\end{align*}.
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