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SLOPE OF A LINE
Learning Area: Mathematics
Curriculum Year: Grade 8
Least Learned Skill: Finding the slope of a line passing through two points.
I. GUIDE CARD
Mountain climbers are eager to reach the peak of the highest mountain in the world.
Americans, Chinese, Japanese, English, and other foreigners have tried climbing into it. Some
has been successful but many of them failed. Even Filipinos tried to conquer the world by
climbing the most dangerous part of the highest mountain in the world, the Mt. Everest.
Dreamers like Romeo Garduce and some other Filipinos once experience the thrill of fulfilling
the dream, reaching the highest peak of the world. Hence, climbers are wondering how steep
they go if they pass the north, the south, the east or the west direction of the mountain.

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In this section, let us explore and find out which direction is more convenient to satisfy
the eagerness of conquering the world by reaching the highest peak in the world, the Mt.
Everest.
II. Activity Card
Definition of Terms:
 Geometrically, every line is a set of points containing at least two different points.
Moreover, any two different points belong to one and only one line. Algebraically, the
line represented by the equation:
Ax + By + C = 0
Where A, B and C are real numbers, and A and B are not equal to zero.
 A point is represented by a dot and shows position on a plane.
 The steepness or inclination of a line is called slope.
Let us begin exploring and find the slope of a line,
but before that let’s have a review on the definition
of a line and a point.

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Activity 1.
Figure 1.
Similarly, to describe the inclination or slope of a line, you choose two points on it to
compute the quotient.
slope = rise = vertical change
run horizontal change
Because the vertical change is moving from one point to another is the difference of the
ordinates and the horizontal change is the corresponding change difference of the abscissa.
Slope = difference of ordinates
difference of abscissa
To find the slope of a line algebraically, consider the equation:
m = y2 – y1
x2 – x1
where:
m – is the slope of the line
y2 – y1 – change in y (difference of ordinates)
x2 – x1 – change in x (difference of abscissa)
To find the steepness of a hill you may determine the vertical rise for every 100 ft of horizontal
run. For example, if a hill rises 20 ft for every 100 ft of horizontal distance, its steepness is the
ratio 20/100 or 20%.
rise = 20 ft 20%
run = 100 ft

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3, 2
4, 5
0
1
2
3
4
5
6
0 1 2 3 4 5
1, 1
5, 3
0
0.5
1
1.5
2
2.5
3
3.5
0 2 4 6
Example 1.
Find the slope of a line passing through
(3, 2) and (4, 5).
Solution:
Let: x1 = 3; x2 = 4; y1 = 2; y2 = 5
Substitute each value on the equation:
m = y2 – y1 = 5 – 2 = 3
x2 – x1 4 – 3
Therefore, the slope of the line passing through
(3, 2) and (4, 5) is 3.
Example 2.
Find the slope of a line through (1, 1) and (5, 3)
Solution:
Let: x1 = 1; x2 = 5; y1 = 1; y2 = 3
Substitute each value on the equation:
m = y2 – y1 = 3 – 1 = 2
x2 – x1 5 – 1 4
Therefore, the slope of the line passing through
(1, 1) and (5, 3) is ½.

5. 5
-4, 4
-1, 2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-5 -4 -3 -2 -1 0
Activity 2.
Another way of checking the slope is counting the number of units on the rise and the
number of units on the run.
Example 3.
m = rise = 2 = _ 2
run -3 3
Counting 3 units from the point going to the
left, the run is -3 and counting 2 units going
up, the rise is +2; therefore the slope is -2/3.
III. Assessment Card
A. Find the slope of the line passing through given points and draw the line.
1. (3, 1); (5,4)
2. (-2, 3); (0, 2)
3. (3, -1); (3, 4)
4. (4, 2); (-3, 2)
5. (2, 4); (-1, -1)

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IV. Enrichment Card
Solve the following:
1. Find the slope of a line passing through (0, 5) and (-6, 8).
2. A ladder is leaning 6 meters away from a building reached 12 meters high from the
ground. Find the inclination of the ladder.
3. Find the slope of a line containing (2, 10) and (-2, 8).

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V. Reference Card
 Slope is the steepness or the inclination of a line. Mathematically, it is the ratio of vertical
change or change in y called rise and horizontal change or change in x known as the run.
Slope = rise = vertical change = change in y
run horizontal change change in x
 The slope of a line passing through two points is determined by the equation:
m = y2 – y1
x2 – x1
where:
m – is the slope of the line
y2 – y1 – change in y (difference of ordinates)
x2 – x1 – change in x (difference of abscissa)