To start the lesson, ask students what they already know about slope. They may know terms such as rate of change and rise over run. Often, students have recollection of these terms but don't remember or understand what they mean or how they relate to slope.
Ask students what it means to have positive or negative
slope. Encourage a student to come to the front of the room and draw a
line with positive slope. Ask classmates if they agree that the line
has positive slope, and then ask how they can tell.
A line with positive slope is pointing upward as you
look to the right. You always want to see if the line is pointing
upward or downward on the right side of the graph, just as we read to the right.
Sketch these two lines with positive slope for students to see.
Ask students to tell you all they can about the two graphs. What's the
same? What's different? Emphasize that although both lines have
positive slope, there is something different about the direction in
which they point. Explain that this description of how slanted a line is can be described by a number called its slope.
Now, draw a third line that has the same slope as the first line, but a different y-intercept. Ask students again for comparisons.
Students should eventually recognize that the third line has the same
slope as the first line. Once they do, they are ready to think about
the slope number as a description of how slanted a line is.
Use the activity sheet for practice and enforcement.
The activity sheet guides students through a process for finding the
slope of a given line. Page 1 is meant to be completed as a class, so
having an overhead slide of this page will be helpful.
Distribute the activity sheets and make sure each student has 1
or 2 colored pencils. Many students enjoy using a colored pencil to
draw and shade the slope triangle, and doing so makes the lesson more
memorable. You might ask students to use one color when they're drawing
the triangle for a line with positive slope, and another color for
triangles representing negative slope.
Shade in the slope triangles with students as shown below.
Encourage students to simplify their fractions on page 1 of the
activity sheet. Point out that for each line, the simplified forms of
the fractions are equivalent — no matter which two points on the line
you student use, or how large the triangle is, you get the correct
On page 2, students are given the slope triangle in the first 3
examples (the top row). In the next 3 examples (middle row), they are
given only the points to use to draw the triangle. In the last 3
examples (bottom row), students have to find the points themselves
before drawing the triangle and determining the slope. The idea here is
to gradually get students comfortable with finding the slope.
While students work on page 2, be sure that they:
- Simplify all fractions
- Determine which lines have negative slope and use a negative fraction to represent the slope of these lines.
This exercise provides students with the skill of finding the
slope of a line from a graph. This skill can be applied to less
abstract examples using real data from a table or a graph.
Questions for Students
1. Which, if any, of the fractions did you have to simplify when you found the slope of a line? How can you avoid the need to simplify a fraction?
2. Suppose you have identified 3 slope triangles for a line to help you find the correct slope. What can you say about the relationship between these triangles?
3. Explain the difference between a line with positive slope and a line with negative slope.
4. Explain the difference between a line with slope 1/2 and a line with slope 2/1.
- Describe what effect using slope triangles had on student understanding of slope.
- What difficulties did students have with this activity?
- How could you extend this activity to rate-of-change problems in the curriculum?
- Describe the effect using colored pencils has on student comprehension of slope.
Students will be able to:
- Determine if the slope of a line is positive or negative
- Express the slope of a line as a fraction
Common Core State Standards – Mathematics
Grade 8, Expression/Equation
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Grade 8, Functions
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.