5.3: Slope
Introduction
Reading Rates
Five of the students in Mrs. Henderson’s class have been tracking the number of books that they have read and have been comparing their results. During the first week, all five finished one book. After the second week, all five had finished two books. During the third week, all five had finished three books. After ten weeks, all five had finished ten books.
What is their rate? If you were to draw the rate of change on a graph comparing books to weeks, what would the graph look like?
Rate of change is one of the things that will be addressed in this lesson. To understand it, you have to understand slope and graphs. All of this will be taught in this lesson. At the end, you will understand how to graph this information.
What You Will Learn
In this lesson, you will learn how to complete the following:
- Recognize the slope of a line as the ratio of the vertical rise to the horizontal run and distinguish between types of slopes.
- Find the slope of a line
- Draw a line with a given slope.
- Model and solve real-world problems by interpreting slopes of simple linear equations as rates.
Teaching Time
I. Recognize the Slope of a Line as the Ratio of Vertical Rise to Horizontal Run, and Distinguish Between Types of Slopes
Have you ever been skiing? Even if you haven't, you probably have an idea about what it might be like to learn to ski.
When someone first learns to ski, he or she usually starts on a slope that is not very steep. Sometimes that slope is called a beginners' slope. After mastering the basic skills of skiing, a person may begin to try slopes that are steeper and more challenging.
In mathematics, the term slope has a similar meaning. In mathematics, the slope of a line describes the steepness of a line. Thinking of a ski slope can help you remember that the slope of a line tells how steep it is.
In mathematical terms, the slope of the line is also referred to as "s", the “rise-over-run.” That is, the slope is the ratio of the vertical (up and down) rise of a line to its horizontal (left to right) run.
To help us understand this ratio, let's look at line \begin{align*}AB\end{align*} on the coordinate plane below.
Imagine placing your finger on point \begin{align*}A\end{align*}. To move from point \begin{align*}A\end{align*} to point \begin{align*}B\end{align*}, your finger would need to move 5 units up and then move 6 units to the right. That is because the line has a rise of 5 units up and a run of 6 units to the right.
So, \begin{align*}\text{slope} = \frac{rise}{run} = \frac{5}{6}\end{align*}.
The slope of line \begin{align*}AB\end{align*} is \begin{align*}\frac{5}{6}\end{align*}.
Line \begin{align*}AB\end{align*} has a slope of \begin{align*}\frac{5}{6}\end{align*}, which is a positive slope.
Notice that line \begin{align*}AB\end{align*} slants up from left to right!
Knowing some basic information about the slope of a line can tell you about its slant.
- A line that slants up from left to right has a positive slope.
- A line that slants down from left to right has a negative slope.
Let’s look at a few examples.
Example
Determine if the slope of each line shown below is positive or negative.
a.
b.
Consider the line in \begin{align*}a\end{align*}.
The line slants down from left to right, so its slope is negative.
Consider the line in \begin{align*}b\end{align*}.
The line slants up from left to right, so its slope is positive.
You should also know about the slopes of horizontal and vertical lines.
- A horizontal line has a run, but does not have a rise.
\begin{align*}\text{slope} = \frac{rise}{run} = \frac{0}{n} = 0.\end{align*}
So, the slope of a horizontal line is zero.
- A vertical line has a rise, but does not have a run.
\begin{align*}\text{slope} = \frac{rise}{run} = \frac{n}{0} = undefined.\end{align*}
Any fraction with a zero in the denominator is undefined. So, the slope of a vertical line is undefined.
Example
Identify the slope of each line shown below.
a.
b.
Consider the line in \begin{align*}a\end{align*}.
The line is vertical, so its slope is undefined.
Consider the line in \begin{align*}b\end{align*}.
The line is horizontal, so its slope is zero.
5G. Lesson Exercises
Answer the following questions about slope.
- What is the slope of a horizontal line?
- What is the slope of a vertical line?
- What is the slope of a line that goes up from left to right?
Take a few minutes to check your answers with a peer.
II. Find the Slope of a Line
Suppose you are given the graph of a line. You can find the slope of that line by choosing two points on the line. Then you can count the units to find the ratio of the rise to the run.
Be sure to consider if the line slants up or down from left to right. This will help you determine if the slope is positive or negative.
Example
Find the slope of line \begin{align*}CD\end{align*} below.
From point \begin{align*}C\end{align*} to point \begin{align*}D\end{align*}, the line rises 6 units up and then runs 2 units to the right. Since the lines slants up from left to right, the slope is positive.
\begin{align*}\text{Slope} = \frac{rise}{run} = \frac{6}{2} = \frac{6 \div 2}{2 \div 2} = \frac{3}{1} = 3.\end{align*}
The slope of line \begin{align*}CD\end{align*} is 3.
Example
Find the slope of line \begin{align*}FG\end{align*} below.
From point \begin{align*}F\end{align*} to point \begin{align*}G\end{align*}, the rise is 4 units down and then 3 units to the right. Since the line slants down from left to right, the slope is negative. You can think of a rise of 4 units down as a negative rise, and represent it as -4.
\begin{align*}\text{Slope} = \frac{rise}{run} = \frac{-4}{3} = - \frac{4}{3}.\end{align*}
The slope of line \begin{align*}FG\end{align*} is \begin{align*}- \frac{4}{3}\end{align*}.
Remember, the slope represents a ratio, not an improper fraction. We cannot say that the slope of line \begin{align*}FG\end{align*} is \begin{align*}-1 \frac{1}{3}\end{align*}.
III. Draw a Line with a Given Slope
We know that we can think of the slope of a line as the ratio \begin{align*}\frac{rise}{run}\end{align*}. We can use this ratio to draw a line with a specific slope.
Example
Draw a line that goes through a point at (–4, –1) and has a slope of \begin{align*}\frac{3}{7}\end{align*}.
Here are the steps to drawing a line with a given slope.
On a coordinate plane, plot a point at (–4, –1). Place the tip of your pencil at that point.
The slope is \begin{align*}\frac{3}{7}\end{align*}. So, \begin{align*}\text{slope} = \frac{rise}{run} = \frac{3}{7}\end{align*}.
The slope is positive, so start at the given coordinates and move your pencil 3 units up and then 7 units to the right to find another point on that same line. Your pencil will end up at (3, 2), so plot a point there. Then draw a line through the two points.
The line you drew passes through a point at (–4, –1) and has a slope of \begin{align*}\frac{3}{7}\end{align*}.
Example
Draw a line that goes passes through (–5, 4) and has a slope of \begin{align*}- \frac{2}{3}\end{align*}.
On a coordinate plane, plot a point at (–5, 4). Place the tip of your pencil at that point. Because it is negative you will either go down two units and to the right three units, or up two units and to the left three units.
The slope is \begin{align*}-\frac{2}{3}\end{align*}. So, \begin{align*}\text{slope} = \frac{rise}{run} = \frac{-2}{3}\end{align*}.
The slope is negative, so move your pencil 2 units down and then 3 units to the right to find another point on that same line. Because it is negative you go down two units and to the right three units, or you could have gone up two units and to the left three units. In our case, your pencil will end up at (-2, 2), so plot a point there. Then draw a line through the two points.
The line you drew passes through a point at (–5, –4) and has a slope of \begin{align*}-\frac{2}{3}\end{align*}.
IV. Model and Solve Real-World Problems by Interpreting Slopes of Simple Linear Equations as Rates
Sometimes, we can use the graph of a straight line to model and solve a real-world problem. In that case, the slope of the line represents a rate of change. We can use what you know about rates and slope to help us work through these problems involving a rate of change.
Example
Becca sells necklaces to make extra money. This graph shows how the total amount she earns for her necklaces increases depending on the number of necklaces she sells.
a. Find the slope of the line shown on the graph.
b. Use the slope to determine how many dollars Becca charges for each necklace she sells.
c. Make a table to show how much Becca earns if she sells 1, 2, 3, or 4 necklaces.
Consider part \begin{align*}a\end{align*}.
Two points on the line are (1, 3) and (2, 6). Starting at the point at (1, 3), the line rises 3 units up and the runs 1 unit to the right.
\begin{align*}\text{Slope} = \frac{rise}{run} = \frac{3}{1} = 3.\end{align*}
Consider part \begin{align*}b\end{align*}.
The slope of the line represents the rate of change. The vertical change (the rise) represents the number of dollars earned and the horizontal change (the run) represents the number of necklaces sold.
So, \begin{align*}\text{slope} = \frac{rise}{run}= \frac{3}{1} = \frac{\$3}{1 \ necklace} = \$3 \ \text{per necklace}\end{align*}.
Consider part \begin{align*}c\end{align*}.
The slope indicates that Becca earns $3 for each necklace she sells. This means that you can multiply the number of necklaces sold by $3 to find the total amount earned.
Number Sold | Total Amount Earned (Dollars) | |
---|---|---|
1 | 3 | \begin{align*}\leftarrow \$3 \times 1 = \$3\end{align*} |
2 | 3 | \begin{align*}\leftarrow \$3 \times 2 = \$6\end{align*} |
3 | 9 | \begin{align*}\leftarrow \$3 \times 3 = \$9\end{align*} |
4 | 12 | \begin{align*}\leftarrow \$3 \times 4 = \$12\end{align*} |
Look back at the table you created for Example 5. It shows that the total amount earned, in dollars, is equal to $3 times the number of necklaces sold. We can use this information to write a linear equation for the problem situation in Example 5. A linear equation is an equation that represents a line.
(total amount earned) \begin{align*}= 3 \ \times\end{align*} (number sold)
\begin{align*}y = 3x\end{align*} where \begin{align*}y = \end{align*} the total amount earned in dollars and \begin{align*}x = \end{align*} the number sold.
In an equation that represents a line, the coefficient of \begin{align*}x\end{align*} represents the slope. The coefficient of a term is the numerical part of the term. For example, in the term \begin{align*}3x\end{align*}, the coefficient is 3. So in the linear equation \begin{align*}y = 3x\end{align*}, the coefficient of \begin{align*}x\end{align*}, which is 3, represents the slope of the line.
When we look at how something changes over time, we can use a linear equation to show a relationship between what is changing and how it is changing. Then a graph gives us a visual display of the change.
Let’s use this information to help us with our introduction problem.
Real-Life Example Completed
Reading Rates
Here is the original problem once again. Reread it and underline any important information.
Five of the students in Mrs. Henderson’s class have been tracking the number of books that they have read and have been comparing their results. During the first week, all five finished one book. After the second week, all five had finished two books. During the third week, all five had finished three books. After ten weeks, all five had finished ten books.
What is their rate? If you were to draw the rate of change on a graph comparing books to weeks, what would the graph look like?
If you read the problem carefully you’ll see that the rate of the students is one book per week.
Let’s list the weeks and the books one student read in a table.
week | books |
---|---|
1 | 1 |
2 | 1 |
3 | 1 |
4 | 1 |
5 | 1 |
6 | 1 |
7 | 1 |
8 | 1 |
9 | 1 |
10 | 1 |
Now, let’s create a graph to show the results. We'll show one line for the rate of book reading, and another for the total number of books read.
You can see that the rate of change is 0. The slope is 0. The line is a horizontal line because the students did not increase or decrease the number of books that the read per week. The numbers remained consistent. The diagonal line is how many books total have been read up to that point.
Vocabulary
These are the vocabulary words used in this lesson.
- Slope
- the slant of a line or the steepness of a line. It is represented on a graph by a ratio of rise over run.
- Rise
- the vertical measurement of a line.
- Run
- the horizontal measurement of a line.
- Positive Slope
- a slope that goes up from left to right.
- Negative Slope
- a slope that goes down from left to right.
- Rate of Change
- the measure of how an event changes over time.
- Linear Equation
- an equation that represents a line.
- Coefficient
- the number found next to the variable.
Technology Integration
Khan Academy Slope and Rate of Change
James Sousa, Determine the Slope Given the Graph of a Line - Positive Slope
James Sousa, Determine the Slope Given the Graph of a Line - Negative Slope
James Sousa, Determine the Slope Given the Graph of a Horizontal and Vertical Line
James Sousa, Example of How to Determine the Slope of a Line Given Two Points on a Line
James Sousa, Another Example of How to Determine the Slope of a Line Given Two Points on a Line
James Sousa, Determine the Slope of a Line Given Two Points on a Horizontal and Vertical Line
Other Videos:
- http://www.mathplayground.com/mv_definition_slope.html – This is a Brightstorm video on defining slope.
Time to Practice
Directions: For problems 1-4, tell if the slope of the line shown is positive, negative, zero, or undefined.
1.
2.
3.
4.
Directions: For problems 5-7, find the slope of each line shown.
5.
6.
7.
8. On the coordinate grid below, draw a line that passes through (–3, 2) and has a slope of \begin{align*}\frac{1}{2}\end{align*}.
9. On the coordinate grid below, draw a line that passes through (-2, 5) and has a slope of -4.
10. Octavius walked at a constant speed while exercising. This equation can be used to show the speed at which he walked.
\begin{align*} y = 4x\end{align*} where \begin{align*}y = \end{align*} the total distance walked in miles, and \begin{align*}x = \end{align*} the number of hours walked.
a. Use the table below to model the total distance Octavius will walk if he walks for 0, 1, 2, 3, or 4 hours at that speed.
Number of Hours \begin{align*}(x)\end{align*} | Total Distance in Miles \begin{align*}(y)\end{align*} |
---|---|
0 | |
1 | |
2 | |
3 | |
4 |
b. Use the grid below to graph a line that models the problem situation. Label the coordinates of at least 4 points on the graph.
c. Determine the speed, in miles per hour, at which Octavius was walking.
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