What you should learn

To find the slope of a line given the coordinates of two points on the line

NCTM Curriculm Standards 2 - 4, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Slope

Rise

Run

Introduction: Alan and Mabel Wong bought a lot in San Francisco on which to build a house. The lot is located on a street having an 11% grade. The length of the sidewalk along the lot is 43 feet. City ordinances state that there must be a 5-foot clearance on each side of the house to the property line. The house they would like to build is 32 feet wide. Will they be able to build their house on the lot they bought? (this problem we will come back to in exercise 4)

What do you think of when you hear the word slope? You might think of how something slants either uphill or downhill. Suppose you placed a ladder against a wall. Then you moved the base of the ladder out about a foot. What happens to the slant of the ladder? Suppose you move it again. What happens?

You should observe that the top of the ladder moves down the wall when the bottom of the ladder moves out. As the top moves down the wall, the slant, or slope, becomes less steep. If you continued to move the ladder out, eventually it would lie flat on the ground, and there would be no slant at all.

The steepness of the line representing the ladder is called the slope of the line. It is defined as the ratio of the rise, or vertical change, tot he run, or horizontal change, as you move from one point on the line to another. The graph below shows a line that passes through the origin and the point at (5, 4) .

slope = change in y (rise)/ change in x (run)

= 4/5

So the slope of this line is 4/5.

 

Definition of Slope: The slope m of a line is the ratio of the change in the y-coordinates to the corresponding change in the x-coordinates.

Exercise 1: Determine the slope of each line.

a.

y change/x change = 3/2

m = 3/2

 

b.

c.

d.

Let's analyze the slopes of the graphs in Exercise 1.

Graph a slopes upward from left to right and has a positive slope

Graph b slopes downward from left to right and has a negative slope

Graph c is a horizontal line and has a slope of 0

Graph d is a vertical line and its slope is undefined

These observations are true of other lines that have the same characteristics.

Since a line is made up of an infinite number of points, you can use any two points on that line to find the slope of the line. So, we can generalize the definition of slope for any two points on the line.

 

Definition Slope Given Two Points: Given the coordinates of two points, (x1, ,y1) and (x2, y2), on a line, the slope m can be found as follows: m = (y2 - y1)/(x2 - x1), where x1x2.

Exercise 2: Determine the slope of the line that passes through (2, -5) and (7, -10).

Let (2, -5) = (x1, y1) and (7, -10) = (x2, y2)

m = (y2 - y1)/(x2 - x1) = [-10 - (-5)]/[7 - 2] = -5/5 = -1

The slope of the line is -1.

 

 

In Exercise 2, it would not matter if you selected (7, -10) for (x1, y1), the slope of the line would still be -1.

You can use a spreadsheet to calculate the slopes of several lines very quickly.

If you know the slope of a line and the coordinates of one of the points on a line, you can find the coordinates of other points on that line.

Exercise 3: Determine the value of r so the line through (r, 6) and (10, -3) has a slope of -3/2.

m = (y2 - y1)/(x2 - x1)

-3/2 = (-3 - 6)/(10 - r)

continue from here...

 

 

You can use slope with other mathematical skills to solve real-world problems like the one presented at the beginning of this lesson.

Exercise 4: Refer to the application at the beginning of the lesson. Will the Wongs be able to build their house on their lot?

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 15 - 39 odd, 40, 41, 43, 45 - 52

Alternative Homework: Enriched: 16 - 38 even, 40 - 52

Extra Practice: Students book page 769 Lesson 6-1

Extra Practice Worksheet: Click Here.

Return to Chapter 6