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## Introduction

Have you ever tried to run up a steep hill?  Ever tried to ride a bike all the way up without walking?  The concept of “steepness” is one that can easily be understood in the real world.  The effort required to get up a steep hill is much greater than that needed to get up a gentle rise.

Judging exactly how steep something is in the real world generally requires some estimation.  You can tell that one hill is steeper than another because you can compare what they look like and how you feel when traveling up or down each one.

In the study of mathematics, the word slope describes the “steepness” of a line.  There are a variety of ways to get an accurate measurement of slope.  In this lesson, you will explore the basic concept of slope.

## Lesson

Slope is most easily measured using a straight line.  This is because a straight line is consistently steep along its entire length.  The concept of slope can be thought of as a “slope ratio” of height compared to length.  The slope of a curvy line is different at various points along the line.  In this lesson, we will be exploring the slope of a straight line.  The slope of a curvy line is a topic that will be introduced much later in your mathematics career.

Measuring slope in the real world may be more difficult because you will often be faced with curvy lines.  Luckily, in mathematics the rules can be set so that you will only deal with straight lines at this time.

The slope of a line can best be described by a ratio.  This ratio is typically given as a comparison of the vertical change in the line compared to the horizontal change in the line.  Here is a formal definition for slope.

The diagram demonstrates the comparison of vertical change and horizontal change.  Basically, the only thing to do is find these two numbers and then insert them into a fraction to get the slope.  Some easier language to learn slope is given below.

Note that the change in y (vertical change) can be called the rise and the change in x (horizontal change) can be called the run.  The slope of the following two lines can be found by simply putting the rise and run into a fraction.

Example #1: The rise is 2 and the run is 10, so the slope can be shown as a simple fraction.

The same process can be used to find the slope of a steeper line.

Example #2: The rise is 6 and the run is 6, so the slope here is also a fraction.  The numerator and denominator cancel out, yielding a (whole number) slope of 1.

The slope in example #2 reduces to the whole number 1.  Notice that this slope has the same rise as it does run.  Additionally, this line goes up at exactly the same rate that it goes to the right.  Geometrically, its slope is a 45° angle.  The number 1 is a defining point for the slope of a line… any line that has a slope greater than 1 is steeper than this line and any line whose slope is less than 1 is not as steep as this line.

In examples 1 and 2, the line rose upwards it went from left to right.  The next example shows what happens when a line falls downward from left to right..

Example 3:  Instead of going up and over, this line goes down and over.  To represent the direction “down”, simply use a negative rise.

So when a line loses value as it goes from left to right (points in a downward direction), the slope is negative.

Special Cases

There are two special cases of lines that have slopes that are neither positive or negative.  These two special lines are horizontal lines and vertical lines.

All movement on a two-dimensional surface can be classified by a combination of vertical and a horizontal movement.  Horizontal and vertical lines are unique because their trajectory can be defined by a single movement on the coordinate plane.    Horizontal lines have a run but no rise, while vertical lines have a rise but no run.

## Review

Find each slope.

1)

2)

3)

4) What is the slope of a horizontal line?

5) What kind of line has a negative slope?

6) Does a curvy line have a consistent slope?

7) Does a straight line have a consistent slope?

1)

2)

3)

4)  A horizontal line has a slope of zero.

5)  A negative slope is a result of a diagonal line that goes down from left to right.

6)  A curvy line does not have a consistent slope.

7)  A straight line does have a consistent slope.