Making connections in geometry

The last few classes have been focused on simple explorations of angle relationships, such as linear pairs, vertical angles, and angles formed by transversals cutting across parallel lines. I continue to reference the parallel parking scenario to motivate where and how these angles come about.

I've had to work in algebraic expressions as values to help prepare the class for questions they may see on common assessments. I have been hard pressed to come up with scenarios that would naturally generate these expressions; it's something I'll need to work on.

For one set of practice problems, students indicated they were getting stuck on some problems. Rather than working through specific situations, I asked students to step back and focus on the angle relationships they are seeing and how they relate to each other. I told them not to worry about the values they were given for different angles. I drew two intersection lines and asked them to tell me the different relationships they saw in the four angles. I then gave a couple of expressions for two angles and asked them how this fit into the situation and could be used. The students that said they were stuck said this helped. I told them to keep working on the problems and we'll discuss them next class.

My purpose is to have students focus on the pieces and how they can be put together to answer questions. By breaking problems down into the components they know and understand, I believe students can piece things back together and become better problem solvers.

Today's class wrapped up with a question tied back to the parallel parking situation. I told them we are turning 45^o in relation to the original position of the parking car. As the car backs up at this angle it forms an angle with the curb going away from the car. I asked what was the size of the angle.

It was interesting that students concluded the angle would be 135^o. I asked them how they knew this. Many stumbled around with rather unconvincing arguments. I asked them to focus on the angle relationships they were seeing. Some students started to recognize that the 45^o angle was a corresponding angle to the angle paired with our angle of interest and that these two angles formed a linear pair. Perfect. Students were using geometry but didn't realize why the angle had to be 135^o. I'm hopeful I can push them further into asking themselves why things work.