Lesson Plan
Lab Edition
Slope Between Two Points
Objectives
Students will:
 describe the meaning of slope.
 investigate horizontal and vertical slopes.
 determine the sign of a slope.
 find the slope between two points.
 plot points that yield a specified slope.
Prerequisite Knowledge
Students are able to:
 plot points on the coordinate plane.
Resources
 This lesson assumes that your classroom has enough computers for all your
students, either working individually or in small groups. For classrooms with
only one computer, see the lecture version of this lesson.
 Rulers, pencil, paper
 Access to http://www.explorelearning.com/
 Copies of the worksheet for each student (optional)
Lesson Preparation
Before conducting this lesson, be sure to read through it thoroughly, and
familiarize yourself with the Defining a line with two points activity on
ExploreLearning.com. You may want to bookmark the activity page for your
students. If you like make, copies of the worksheet for each student.
Lesson
Motivation:
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Break students up into small groups of two or three. Give each group a sheet of
paper with a right triangle on it. Make the horizontal leg of the triangle 12cm.
Make the vertical leg of the triangle 1cm.
1cm
12cm
Explain to the students that in order for some buildings to be handicap
accessible, they must have an entrance ramp. The triangle represents a scale
model of the steepest ramp allowed by law.
Have students construct triangles with the following dimensions.
4cm
1
12cm
1cm
Get the students to explain why ramps similar to these may not be suitable for
handicapped access. Try to lead the students towards words like “rise” and “run.”
Have students construct triangles with the following dimensions.
3 cm
0.5cm
x cm
x cm
Ask the students to find the maximum value for x allowed by law for a handicap
ramp. Have students explain the methods used to get their answers.
Other applications of slope
Ask students to name other areas where slope is important. A few applications
are outlined below.
Construction
Most houses have roofs that are sloped. Builders use the word ‘pitch’ to describe
the slope of a roof. For example, a roof with a 5:12 pitch would rise 5ft. for every
12ft. of run.
Road Signs
On many mountainous roads, trucks are warned of steep sections of decent.
Most of the warnings are in the form of signs. For example, a sign may read,
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”Trucks use lower gears. 8% grade for the next 2 miles.” This means the road
drops 8 ft for every 100 ft of horizontal distance. If trucks do not take the right
precautions, their brakes may fail before they reach the bottom of the mountain.
The “Slope between two points” activity
In your words, go over the information below with the students before moving on
to horizontal slope.
The slope between two points is determined by dividing the “ rise” by the “run.”
“Rise” is the vertical distance between the points and “run” is the horizontal
distance between the two points. Say we have two points (x1, y1) and (x2, y2).
The vertical distance is y2 - y1 and the horizontal distance is x2 - x1. Slope is
usually denoted by the letter m. The formula for slope is m = (y2 - y1)/(x2 - x1).
(x2, y2).
(x1, y1)
Rise = y2 - y1
Slope = m = (y2 - y1)/(x2 - x1)
Run = x2 - x1
To see this in action, go to the Defining a line with two points activity at
ExploreLearning.com. If you are using a worksheet, you can give it to your
students now if you want them to follow along, or you can hand it out at the end
of class for them to review later.
Horizontal slope
Have students plot (x1, y1) and (x2, y2) on the plane in a horizontal relationship.
Now have them select the “compute slope box.”
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Ask students to state the slope of their line.
Have students drag the x2 bar to move the (x2, y2) point horizontally. Ask the
students to explain what they observe as they move (x2, y2) horizontally.
Students should notice that the slope between any two horizontal points is zero.
Positive/Negative Slope
Have Students move (x1, y1) and (x2, y2) to the points (0,0) and (4,0).
This time, have them drag the y2 bar to raise the right side point up and down.
Ask the students what the sign of the slope is when the right side point is higher
than the left side point. Now ask the students what the sign of the slope is when
the right side point is lower than the left side point. Have students come up with
conjectures about when a slope is positive or negative. Students should be able
to see that when the left side point is higher than the right side point, the slope is
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negative, and when the left side point is lower than the right side point, the slope
is positive.
Undefined Slope
Summarize the rules for positive, negative, and zero slopes. Have students
place (x1, y1) and (x2, y2) on the plane in a vertical relationship.
Ask students to state the slope of the vertical line. Now have them drag the y2
bar to move (x2, y2) up and down. Ask students what they observe as they move
(x2, y2) vertically. Students should observe that the slope between any two
vertical points is undefined.
Determining Slope Graphically
Students should start by deselecting the “compute slope” box. Have them place
(x1, y1) at the origin and (x2, y2) at the point (4, 6).
Remind students that slope is rise/run. Have students determine the rise
between the points by counting the units that lie vertically between (x1, y1) and
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(x2, y2). Have students determine the run between the points by counting the
units that lie horizontally between (x1, y1) and (x2, y2). Once students find the
correct rise and run, have them calculate the slope.
Students should now check the “compute slope” box to see if their answers are
correct.
Repeat this section with the several sets of points.
Finding a Second Point Given a Point and a Slope
Have students graph a line contains the point (-4,2) and has a slope of –2/3.
Ask students how they could find a second point on the line given only one point
and a slope. Remind them that slope is rise/run. Ask them what the run is for
this line. Ask them how many spaces to the right of (x1, y1) will be (x2, y2). Ask
them what the rise is for this line. Ask them how many spaces below (x1, y1) will
be (x2, y2). Have the students give you the coordinates for a second point.
Students should now type the coordinates in for (x2, y2). Have them check the
‘compute slope’ box to see if the slope between these two points matches the
slope of the given line.
If it does, the students have found a second point on the given line.
Repeat this exercise with other points and slopes. Have students find multiple
points on a given line.
Conclusion
Slope is defined as rise/run. Horizontal lines have slope m=0. Vertical lines have
an undefined slope. Lines that move upward left to right have positive slope, and
lines that move downward from left to right have negative slope. Slope can be
determined using the formula: m = (y2 - y1)/(x2 - x1). In subsequent lessons we will
use the concept of slope to help us determine the equations of lines.
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