Back To CourseSupplemental Math: Study Aid
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Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.
The slope of a line is the ratio of the amount that y increases as x increases some amount. Slope tells you how steep a line is, or how much y increases as x increases. The slope is constant (the same) anywhere on the line.
One way to think of the slope of a line is by imagining a roof or a ski slope. Both roofs and ski slopes can be very steep or quite flat. In fact, both ski slopes and roofs, like lines, can be perfectly flat (horizontal). You would never find a ski slope or a roof that was perfectly vertical, but a line might be.
We can usually visually tell which ski slope is steeper than another. Clearly, the three ski slopes get gradually steeper.
In mathematics, we often want to measure the steepness. You can tell that slopes B and C are higher than slope A. They are both seven units high, while slope A is only four units high. So, it appears that height has something to do with steepness.
Slope C, however, is clearly steeper than slope B, even though both are seven high. So, there must be more to steepness than height. If you look at the width of slopes B and C, you see that slope B is ten units, while slope C is only six units. The narrower ski slope is steeper.
It is not height alone or width alone that determines how steep the ski slope is. It is the combination of the two. In fact, the ratio of the height to the width (the height divided by the width) tells you the slope.
Think of it this way: suppose you need to change seven feet in height to get from the bottom of the ski slope to the top. For the moment we will pretend you are trying to climb up to the top of the slope. We will discuss going downward later. If you have only four feet straight ahead of you (the width in the picture) in which to get to the top, you have to climb up at a very steep angle. If, on the other hand, you have six feet ahead of you in which to ascend those seven feet, the angle is less steep. It is the relationship between height and width that matter.
You could write the relationship like this:
Slope = (Change in height)/(Change in width)
Slope = rise/run
If y represents the vertical direction on a graph, and x represents the horizontal direction, then this formula becomes:
Slope = (Change in y)/(Change in x)
In this equation, m represents the slope. The small triangles are read 'delta' and they are Greek letters that mean 'change.'
For the first ski slope example, the skier travels four units vertically and ten units horizontally. So, the first slope is m = 4/10.
The second ski slope involves a seven unit change vertically and ten units horizontally. So, the slope is m = 7/10. The second slope is steeper than the first because 7/10 is greater 4/10.
The third ski slope involves a seven unit change vertically and six units horizontally. So, the slope is m = 7/6. The third slope is the steepest of all.
The Cartesian plane is a two-dimensional mathematical graph. When graphing on it, a line may not start at zero as in the ski slope examples. In fact, a line goes on forever at both ends. The slope of a line, however, is exactly the same everywhere on the line. So, you can choose any starting and ending point on the line to help you find its slope. It is also possible that you might be given a line segment, which is a section of a line that has a beginning and an end. Or, you might be given two points and you are expected to draw (or imagine) the line segment between them. In all these situations, finding the slope works the same way.
Just like with the ski slope, the goal is to find the change in height and the change in width. For the line segment in the image, you can simply count the squares on the grid.
The difference in height between the two points is three units (three squares). The difference in width between the two points is two units (two squares). So, the slope of the line segment (the slope between the two points) is m = 3/2.
In mathematics class, you may memorize a formula to help you get the slope. The formula looks like this:
This formula is really the same thing as we used before. The top says to take the two y-values and subtract them. The bottom says to take the two x-values and subtract them. There is one important key: subtract them in the same order both times. That means that if you use the y-value from the point further to the right first in the formula, then use the x-value from the point furthest to the right first.
For example, in the graph, you would put the x and y values into the formula like this: m = (5 - 2) / (4 - 2) = 3/2
One more thing to understand is that when you plug numbers into the slope formula, you could get a negative number out. For example, suppose you have the two points: (3, 2) and (1, 4) as shown in the picture.
When you put them into the slope formula you get m = (3 - 1) / (2 - 4) = 2 / -2 = -1. Or, if you put the numbers in in the opposite order (which is fine), you get m = (1 - 3) / (4 - 2) = -2 / 2 = -1.
The slope of the line is negative! What does that mean? Well, negative slope means that the line is slanting downward from the left to the right.
When you think of a ski slope, you tend to think of traveling downward because you ski downward no matter which way the slope is facing. However, in math, you always imagine you travel left to right, just as you do with your eyes when you read. So, the ski slopes on this page are all positive slopes. A negative sloping ski slope would have you skiing downward from left to right.
In summary, a slope is simply a way of measuring how two points differ in height (vertical distance) relative to width (horizontal distance) as you move from left to right between them. You may wish to simply think of slope = rise/run. The slope is a number that tells you how much the line 'rises' (increases in the y-direction) as it 'runs' (increases in the x-direction).
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Back To CourseSupplemental Math: Study Aid
1 chapters | 19 lessons