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Slope of a Line Prepared by Gladys G. Poma Concept : The slope of a straight line is a number that indicates the steepness of the line. The slope tells us how much the line rises from one point to another located one unit to the right. Rise 1 unit Examples : 1 unit Slope = 2 ½ or 0.5 of a unit 1 unit 2 units 1 unit Slope = 1 1 unit Slope = ½ 0r 0.5 2 EXERCISES : 1) A. Using only the concept of slope and a ruler, find the slope of the following lines: Slope = Slope = B.Using only the concept of slope and a ruler draw lines with the following slopes: Slope = 3 Slope = ¼ 0r 0.25 3 2) A. Using the concept of slope and the grid find the slope of the following lines: Slope = Slope = B. Using the concept of slope and the grid draw lines with the following slopes: Slope = 2 Slope = 2.5 4 + _ Positive: When the line actually rises or goes up. Negative: When instead of rising, the line goes down. Example: Positive Slope 2 units going up Moving left to Right Negative Slope 1 unit 2 units going down or -2 1 unit Slope = 2 or +2 Slope = - 2 5 EXERCISES : 1. Indicate the sign of 2. Find the slope of the slope for each line shown below. the following lines. 3. What is the slope of the line in the graph? Choose the best answer. Slope = Use a ruler Slope = a) b) c) d) e) ½ 2 -2 -½ 1/3 6 The slope indicates rise per unit of horizontal right movement and this value is the same everywhere along the line, because the line is straight. Then, if we move more than one unit to the right, the rise will be proportional. That is why, to find the slope we can use any two points on the line and find the ratio of their vertical distance to their horizontal distance. Y2 Y1 1 unit X2 X1 Definition : Slope Vertical Distance y = = = Horizontal Distance x Then, there are two ways to find the slope of a line: Use a graph to find Use the formula. x and y Y2 –Y1 X2 – X1 GED Formula and find the ratio; or 7 Steps: y 1.- Use the graph to choose a x and a y with lengths that have an exact number of units. 2.- Slope = x y/ y Example : In the graph: x x= 4 and y = 5, then the = Slope = x 5 4 8 EXERCISES: (From the book: GED Mathematics . Steck-Vaughn) Find the slope of each line 1. y y 2. 3. y x x x Slope = Slope = y 4. Slope = 5. The line that passes through the points: (1,-3) and (0,1). y Note: Draw the line and find the slope using the graph x x Slope = Slope = 9 Y2 Point Formula : 2 Point 1 = (X1,Y1) Y1 Point Y2 –Y1 X2 – X1 Point 2 = (X2,Y2) 1 If the coordinates of two points are given, we do not need the graph to use the Slope Formula. X1 Example : X2 In the graph : Point 1 = ( 2 , 4 ) and Point 2 = ( 9 , 8 ) X1 Y1 Then, Slope = Y2 –Y1 X2 – X1 X2 Y2 8 – 4 = 4 = 9 – 2 7 10 EXERCISES: (From the book: GED Mathematics . Steck-Vaughn) Find the slope of the line that passes through each pair of points. 1. (4,5) and (3,- 4) 4. The points are shown in the graph y x 2. (- 3,- 3) and (- 2,0) Use the formula to solve. Suppose you do not know the concept of a slope and the graph method, only the formula. 3. (- 4,3) and (5,3) 11 Objective : Compare the two given methods to find the slope: Graph and Formula . The student will work with both methods and choose which one they like better. 1) Two teams : Both teams solve the same exercise, but each team uses a different method. Students work in pairs or independently. The student who finishes first in each team writes the solution on the board. The team/method that finishes first wins. Data: Coordinates of two points 2) Same as part 1, but switch methods between teams. Data: Coordinates of two points 3) Similar to part 1, but now each person chooses his or her favorite method . If everybody chooses the same method, then the students that finish first in parts 1 and 2 must use the other method. Data: Coordinates of two points At the end, discuss which method won more times and why. 12 The slope of any horizontal line is 0. A vertical line has no slope. All lines with the same slope are parallel. If we have the equation of a line written in the form: y = mx + b, where m and b are numbers, then m is the slope of the line. Examples : 1) The line with equation y = 3x - 4 has slope 3. 2) The line with equation y = -x + 5 has slope -1. 13