The Right Triangle

Math Content: Exploring Rise over Run
Instructional Strategy: Group Presentation, Questioning

The Mathematics Framework for California Public Schools points out that "the fact that the slope of a line is the same regardless of which pair of points on the line are used for its definition depends on the considerations of similar triangles." [Source: California Department of Education).

But what about graphs that do not form lines? Their slopes actually change between pairs of points, so the slope on different parts of the graph are different, depending upon which pair of points are used. Let's use a graph of gravitational acceleration as an example. An object dropped from a bridge will travel 16 feet during the first second. But, because of gravitational acceleration, the object will travel an additional 48 feet during the next second. So, rate of change (and thus, the slope) in this nonlinear situation depends upon where on the graph you are.

In the graph of a linear relationship, you can see this constant relationship of the rise over the run in a series of right triangles that show the relationship between every set of points on the line.

As you observe the following classroom clip, think about the relevance of this triangle to the equation of the line. What do you notice about the students? poster?

Classroom Clip:

  • What questions might have been asked about the slope triangle in this graph?
  • What other questions might have been asked of this group?
  • What are the implications for your work with students?