The slope of the line joining two distinct points is calculated by finding the change in y -values and dividing by the change in x -values. For example, the slope between the points (7, -15) and (-8, 22) can be computed as follows:

- The difference in the
y -values is −15 − 22 = −37. - The difference in the
x -values is 7 − (−8) = 15. - Dividing these two differences, we find that the slope is
−3715

The points A = (2, 3), B =(18,11) and C = (18,3) define three different lines: AB←→ , BC←→ , and AC←→

- One of the three lines has a slope of zero and one of the lines has an undefined slope. Determine which of the lines has slope zero and which has an undefined slope.
- Create two points,
D andE (different fromA ,B , andC ), that define a line whose slope is zero. State their coordinates and graph them on the coordinate grid along with lineDE←→ . - Create two points,
F andG (different from the preceding points), so that the line joiningF andG has undefined slope. State their coordinates and graph them along with lineFG←→ on the coordinate grid. - Describe the characteristics, in terms of coordinates and graphs, of lines that have a slope of zero.
- Describe the characteristics, in terms of coordinates and graphs, of lines that will always have an undefined slope.

##
**Commentary**

The "change in y divided by the change in x" can be computed for any two points in the plane... unless the points have the same x-coordinate. The purpose of this task is to help students understand

*why*the calculated slope will be zero for any horizontal line and undefined for any vertical line.
This task is based on Slopes Between Points on a Line. In that task, students argue that for any two points on a line, the "slope triangles" have to be similar, and as a result the lengths of the sides of the triangles will be proportional. This is why the slope between any two points on a particular line will always be equal, and why we talk about "the" slope of a line. This task investigates a degenerate case where "slope triangles" can not be constructed, since the line is parallel to one of the axes.

This instructional task is intended to be used in a class discussion. Students can work on part (a) independently or in small groups. The class could discuss their answers before proceeding with the remaining parts. Students should be given a chance to try to create descriptions for part (d) and (e) on their own. Some may notice the graphical requirements, and some may notice the requirements of the coordinates. Both of these should be discussed and understood by the whole class, and the ideas should be connected to the slope-calculating procedure introduced at the beginning of the task.

## Solution

Another possible response: Looking at the graph of

c. Sample Response

d. Lines whose slope is zero are horizontal. Two points on the line must have the same y-coordinate.

e. Lines with an undefined slope are vertical. Two points on the line must have the same x-coordinate.