I used the MAP Lesson Finding Equations of Parallel and Perpendicular Lines as a start to our unit on Coordinate Geometry.
One of the interesting tasks was for students to determine the equations of three other sides of a rectangle given that one side had the equation y=2x+3. I would have never thought to leave off the x- and y-axes had I created the question, but it brought about some interesting responses from my students for us to discuss.
We ran out of time to discuss the problem during the first lesson (imagine that!), so I asked for students to consider the problem outside of class. During the second class, I collected their work, but we did not go over it. The following are some of the responses that I got.
I was very surprised by the progression of responses. It would have never occurred to me that students would have suggested the same line for the two opposite sides of a parallelogram. Several students remembered the slope criteria for perpendicular lines from their algebra class, but, again, more than one used the same line for the other pair of opposite sides of the rectangle.
I kind of liked the generalizing that several students did for the y-intercept of the equations of the sides, but they were not totally prepared for the practice of “reason abstractly and quantitatively”. Did they intend for their values of b or r (see above) to be equal? They used the same constant to represent the values.
The day after I looked at the student work, we revisited the problem as a class. We started with a Graphs page. Only a few of the students that I have this year used TI-Nspire handhelds last year in algebra, so their experience with graphing on TI-Nspire was limited. In fact, most had not thought of graphing the equations as an option. They graphed the given equation. I showed them the first picture above. Could the same equation work for the opposite side? Some of them had to graph the equation again to realize that it wouldn’t work; others knew immediately.
Students determined a second equation for the opposite side. I used the Class Capture feature of TI-Nspire Navigator to monitor students’ progress – I am able to set Class Capture to refresh every 30 seconds so that I can walk around and discuss the problem with individuals or groups who need help.
As students continued to work, I was pleased to see them troubleshooting their own work. They entered the practice of “attend to precision” on their own – I didn’t have to tell them that lines with slopes of 2 and -2 are not perpendicular.
I didn’t have to tell them that y=-½+3 is the equation of a horizontal line instead of an oblique line. The technology provided them the opportunity to figure that out for themselves.
Even if I ask a similar question on a no calculator assessment later, I want my students learning and making sense of the mathematics with the technology.
And speaking of a future assessment, I’ve been trying to think of a good question to follow up on this task. What if I give them the equation of one side of a square and ask for the rest?
Or the equation of one side of an isosceles trapezoid and ask for the rest? If I remember correctly, what happens with the slopes of the two legs is interesting.
At least I’ll never run out of problems as the journey continues…