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## Thursday, February 4, 2010

### Comparing Slopes of Perpendicular Lines

This is a particular concept in 8th Grade Algebra that I find difficult to teach. From my perspective, it can easily fall into the category of "memorization" rather than conceptual understanding. The CPM text we use has a pretty nice way of leading the students towards a "discovery" of this relationship. They draw a pair of perpendicular lines on transparency sheets and then place the lines on graph paper to determine their respective slopes. The students collect data in a table and then analyze it.

Perpendicular lines are complicated. If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will be a decreasing line). So perpendicular slopes have opposite signs. The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down

I find that the "opposite" nature of slope (positive or negative growth) is not really a problem. But determining the slope accurately can be problematic. Many students are inaccurate or wishful in determining the slope of the lines. They lack a certain ease with estimation: either they over generalize or they are nit picky about when a line crosses a lattice point on the graph paper. This can lead to them not really "discovering" the reciprocal nature of the slope ratio.

When they make this error, I usually ask them to show me the orientation of the lines on the graph paper. At that point, if their error is not self evident to them, I find myself telling them how to see the slope correctly. I am cautious with this final step because it tends to take away some of the "discovery" nature of the activity and disempowers the student. But if I never intervene, they may never really figure out the lesson in the first place.

This year, with the addition of the SmartBoard, the lesson was far easier to present and analyze. I can make far crisper and interesting lines and graphs and manipulate them with grace and meaning. Thanks to this technology, many more of my students were able to "get" the opposite reciprocal concept easier. I was able to save the graph and put it up on the school website as well. This has been an important addition to the conceptual teaching I strive for.

Perpendicular lines are a bit more complicated. If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will be a decreasing line). So perpendicular slopes have opposite signs. The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down