Recently, a high school teacher asked me about the idea related to slopes of perpendicular lines. As the conversation went on, it became clear to us that we don’t just want students to “know” that the slopes of perpendicular lines are negative reciprocals. Rather, we want students to be able to make sense of relationships between lines. Certainly, we could tell students about this relationship…we could ask them to memorize it…we could check to see if they memorized this bit of trivia by putting some item on the next test. We decided that it would be better to allow students the opportunity to try to make sense of this idea and that this could be done in class in small groups or even as a HW assignment – assuming that students have previously worked to make sense of the idea of slope. Here is a first draft of an activity/assignment. Notice the quest to guide students towards making sense and then demonstrating the sense that they have made by preparing a convincing argument. Imagine creating a classroom where students are engaged in making sense and communicating their thinking to one another. Any ideas for improving this?

1. Find the slope of the line in the graph below.

2. We now rotating the line 90^{o} about point *B*. Note that the marks on the line segments are to help you to make the connections between the original line and the rotated line. Find the slope of the rotated line.

3. Describe the relationship between the slope of the original (black) line and the rotated (red) line. Do not focus on formulas but rather on the geometry seen in the graphs.

4. Confirm or adjust your ideas from number 3 by sketching the graphs of linear functions and finding the slopes of these lines. Then, rotate the line about a point and find the slope of the new line.

5. Prepare a convincing argument that describes, in general, the relationship between the slopes of perpendicular lines. Use graphs, formulas, and words to help make your argument. Be prepared to present your argument in class tomorrow.

Scott

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This is an excellent activity. I like the geometric approach which makes this idea so intuitive.