Strategic Intervention Materials in Mathematics I (Slope of a Line)

SLOPE OF A LINE Learning Area: Curriculum Year: Proponent/Writer: School and Address: Least Learned Skill: Mathematics I First Year Jayson O. Daguro O¶Donnell High School Annex Sta. Juliana, Capas, Tarlac 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. 1 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 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. 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. 2 Activity 1. 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%. Figure 1. rise = 20 ft 20% run = 100 ft 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) 3 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. 6 5 4, 5 4 3 2 3, 2 1 0 0 1 2 3 4 5 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. 3.5 3 5, 3 2.5 2 1.5 1 1, 1 0.5 0 0 1 2 3 4 5 6 4 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. 4.5 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. -4, 4 4 3.5 3 2.5 -1, 2 2 1.5 1 0.5 0 -5 -4 -3 -2 -1 0 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) 5 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). 6 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 run = vertical change = horizontal change change in y 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) 7