A Look at Slope

You already know that math is used in many different ways. One way we often use math is determining rate of change. Rate of change is a ratio that compares the changes in two related values. Common examples are Miles Per Hour, Miles Per Gallon, Price Per Pound, and many many more. Each of these examples are a specific type of rate of change called "unit rate." Unit rate is a rate of change where the denominator is always over a change of 1. For example, 50 Miles Per Hour is the comparison of change in distance of 50 miles to a change in time of 1 hour. We often calculate unit rate from a more general rate of change: "You traveled 150 miles in 3 hours" compares your change in distance by 150 miles to the change in time of 3 hours. Simplifying the fraction of 150/3 gives the unit rate of 50MPH.

As we begin looking at algebra through graphing, we are going to begin to learn how rate of change affects what is called the slope of a line.

Slope can be easiest defined as the "steepness" of a line. It gives a sense of direction for the line. The steeper a graph's line, the greater the slope.

We often draw graphs that show the comparison between two variables. The y axis represents the numerator of the ratio where the x represents the denominator of the ratio. Below are the graphs of a car, a rocket, and a bicycle. Each graph is comparing the rate of change in distance to the change in time. Can you tell which graph belong to each vehicle?

In math, we often use the letter 'm' to represent slope. Below are several varying slopes and their graphs. Notice that the bigger the m value gets, the steeper the graph gets.

All lines can be defined as having one of four types of slope. The four types of slope are:

Positive Slope - When a line increases in height as it travels left to right

Negative Slope - When a line decreases in height as it travels left to right

Zero Slope - When a line maintains its height as it travels left to right (Horizontal Lines)

Undefined Slope - A line that is straight up and down (Vertical Lines)

In all three of the above graphs, the lines have a positive slope.
Below are examples of the three other types of slope.

We often find m in the equation of a line. It's easiest to find m's value in the Slope-Intercept form of an equation. Slope-Intercept form looks like this:

y = mx + b

In this form, m and b will be written as numbers, while y and x are left variables.
Here're examples of what an equation in Slope-Intercept might look like:

y = 3x + 5
y = -4x + 10
y = 7x - 2

In the first equation, 3 is the m value. In the second equation, -4 is the m value. In the thrid, 7 is the m value. These are the values you would give as the slope. Take note that m is always paired with x. Sometimes the equation may change the order of the mx and b.
(y = b + mx)

Can you tell me the slope of the lines with these equation?
(Remember that m is always paired with the x)

y = 10x - 3
y = 7 - 4x

We said earlier that rate of change affects the slope of our graphs; this was a bit misleading. Rate of change doesn't really affect slope; Rate of change is slope.

Take speed for instance. If at 3pm I enter the highway at mile-marker 25, and at 5pm I exit the highway at mile-marker 85, I can find my average rate of change by comparing the change in the two values, distance and time. My distance changed by 60 miles and the time changed by 2 hours (60/2). If I reduce that to the unit rate, I would find that I had been traveling at a speed of 30MPH. If I graphed my trip on the Cartesian Plane, I would also find that my slope is 30. Lastly, if I had an equation that generated my distance, I would find that m=30.

So, we can find rate of change by looking at the slope of a graph or by finding the m value in an equation. I can also find slope by finding the m value of an equation or by finding rate of change. All three are the same.

In algebra, we use the following formula to find slope and rate of change:

To use this equation, separate the data by collection, collection 1 and collection 2. The two different data types are sorted x and y. Input the values into the formula and solve for m.

For our example of finding our average speed, the first data values collected were 3pm and mile-marker 25. The second data values collected were 5pm and mile-marker 85. We'll separate these into two groups, (3, 25) & (5, 85). Since (3, 25) was collected first, we'll call it group 1, and (5, 85) group 2. We usually want to use time as an x value; for this reason, put time values first in the ordered pair (x, y).

Now that we've organized our data, we can plug the values into our formula...

and solve for m.

Now that we get m = 30, we can say that we were traveling at a rate of 30 miles per hour.

For information on finding Rate of Change using graphing tools, click here!