1.7: Words that Describe Patterns
Many bicyclists have biking watches that are able to record the time spent biking and the distance traveled. They are able to download the data recorded by their watches to a computer and view a table with times in minutes in one column and distances in miles in another column. How could they use this data to write an English sentence or an algebraic expression?
Using Words to Describe Patterns
Sometimes patterns are given in tabular format (meaning presented in a table). An important job of data analysts is to describe a pattern so others can understand it.
Let's describe the following patterns in words:
- \begin{align*}&& x && -1 && 0 && 1 && 2 && 3 && 4\\
&& y && -5 && 0 && 5 && 10 && 15 && 20\end{align*}
xy−1−50015210315420
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.”
- Zarina has a $100 gift card and has been spending money in small regular amounts. She checks the balance on the card at the end of every week and records the balance in the following table. Using the table, describe the pattern in words and in an expression.
Week # |
Balance ($) |
---|---|
1 | 78 |
2 | 56 |
3 | 34 |
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.” As we saw in the last lesson, this sentence can be translated into the algebraic expression \begin{align*}100-22w\end{align*}
Using Expressions
Note that the expression found in the second problem can be used to answer questions and predict the future. Suppose, for instance, that Zarina wanted to know how much she would have on her gift card after 4 weeks if she used it at the same rate. By substituting the number 4 for the variable \begin{align*}w\end{align*}, it can be determined that Zarina would have $12 left on her gift card.
\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 would have $12 left on her gift card.
Examples
Example 1
Earlier, you were told that bicyclists are able to download the their data on the time they spent biking and the distance they traveled. This data can be represented as a table with the times in minutes in one column and the distances in miles in another column. How can the bicyclists use this data to write an English sentence or an algebraic expression?
As shown in this section, tables are useful for finding patterns. To write a sentence or expression about the data, the bicyclists must look at the data and find a pattern for how the distance changes as time increases.
Example 2
Jose starts training to be a runner. When he starts, he can run 3 miles per hour. After 5 weeks of training, Jose can run faster. After each week, he records his average speed while running. He summarizes this information in the following table:
Week # |
Average Speed (miles per hour) |
---|---|
1 | 3.25 |
2 | 3.5 |
3 | 3.75 |
4 | 4.0 |
5 | 4.25 |
Write an expression for Jose's increased speed and predict how fast he will be able to run after 6 weeks.
We will use \begin{align*}w\end{align*} to represent the number or weeks. Jose's speed starts at 3 mph, and from the table we can see that it increases by 0.25 miles per hour every week. This gives us the expression \begin{align*}3+0.25w\end{align*}. Now we substitute in \begin{align*}w=6\end{align*} and get the following:
\begin{align*}3+0.25(6)\end{align*}
\begin{align*}3+1.5=4.5\end{align*}
If Jose keeps up his training, by the end of the 6th week, he should be able to run 4.5 miles per hour.
Review
In questions 1–3, 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 && 3 && 4 && 5 && 6\\ &\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*}.
Review (Answers)
To see the Review answers, open this PDF file and look for section 1.7.
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tabular
Being represented in a table.constant
A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a.Equation
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.greater than
The greater than symbol, , indicates that the value on the left side of the symbol is greater than the value on the right.greater than or equal to
The greater than or equal to symbol, , indicates that the value on the left side of the symbol is greater than or equal to the value on the right.inequality
An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are , , , and .less than
The less-than symbol "<" indicates that the value on the left side of the symbol is lesser than the value on the right.less than or equal to
The less-than-or-equal-to symbol "" indicates that the value on the left side of the symbol is lesser than or equal to the value on the right.not equal to
The "not equal to" symbol, , indicates that the value on the left side of the symbol is not equal to the value on the right.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.Image Attributions
Here you'll learn how to interpret the figures in a table by writing an English sentence or an algebraic expression. You'll also use the algebraic expressions you find to make predictions about the future.
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