Written by Jonathan Osters *@CallMeJosters, *The Blake School

A few weeks ago, we began the journey of parallel lines in geometry. I have taught this course for several years now, and the first day of this unit is typically excruciating for me – a lot of bland definitions that don’t really have any connection to anything interesting. This year, I decided to change things up and have students create the definitions and conjectures on their own, but I would need a means to do that.

I have seen a number of posts from the wonderfully knowledgeable teachers of #MTBoS (Math Twitter Blogosphere) on Twitter, and many of them had been beginning lessons asking students the questions, “What do you notice?” and “What do you wonder?” I have been skeptical about trying this in a high school level class, but I decided to give it a try anyway.

I used the following figure from Sketchpad and projected it on the board. Then I gave students a few minutes to notice and wonder (My only directions were that the “Notice” statements had to be statements, and the “Wonder” statements had to be questions).

The following were some of the “Notice” and “Wonder” statements I got throughout the day in my three sections of Geometry:

Then I asked if there was anything among the things we noticed that allowed us to answer any of the questions that we wonder. Some, like the distances, were directly measurable in Sketchpad, while “are the lines parallel?” required a different set of skills. These students have taken Algebra 1, and several pointed out that if these lines had been parallel, we would have been able to determine if they were parallel. I obliged them, despite the fact that discussions of slope weren’t an official part of the lesson.

So from here, we were able to re-familiarize ourselves with slope as a number that measures steepness, and that if two different lines have the same steepness, they will be parallel. At this point, many students were at least able to answer the question, as 6.63-1.131.87-(-14.08)=0.345, while 0.78-(-5.40)5.19-(-13.51)=0.330, so the lines are not parallel.

I asked if there was anything they could notice and wonder now that there is grid on the board. Several students wondered if the lines were making the same angle with the y-axis. I removed the grid and just placed a vertical line where the y-axis had been, and measured the angles.

Many students then made the argument that had the two angles been equal in measure, the lines would have been parallel. They also argued that they could find the measures of the other six angles, and that if they could determine if “the angle at any corner” were congruent “to the angle at the same corner on the other line,” then the lines would be parallel. Some noticed that it would be enough to know that the “two angles on the inside but the other side of that axis” were congruent. At this point, it was they who decided we would benefit from some terms for such angles. And voila, the definitions for Alternate Interior Angles, Corresponding Angles, and the like. We were able to modify their original conjecture to: If Corresponding Angles on two lines are congruent, then those lines are parallel. This is *exactly* what I had hoped they would see, and while we ran out of time to prove the conjecture, we would begin there the next day, and the lesson was so much the better for it.

Noticing and wondering was a very powerful technique, especially when introducing something brand new. I will be utilizing this strategy frequently. Thanks to the teachers of #MTBoS for helping me learn something new!