Parallelogram properties - an initial investigation

Today we dove into looking at parallelograms. To start things off, I had students grab small whiteboards that have an xy-grid on them, dry erase markers, and a straight edge. On the board, I projected a grid system and drew two pairs of parallel lines that intersected with each other and labeled the intersection points as A, B, C, and D.

I asked students to draw two pairs of intersecting parallel lines on their boards and label the intersection points. I noted that we wanted to focus on the figure defined by the intersection points, the parallelogram ABCD.

The task was for students to use their knowledge of parallel lines and intersections with transversals to identify everything they could about the properties of the parallelogram. I told them to reference their notes if they couldn't remember the different properties and theorems we had previously worked with.

As I walked around the room, students had their figures drawn and were actually remembering or recovering from their notes different facts and theorems they could use. I talked with different groups, asking what they had found. Sometimes, I had to redirect the focus to the actual figure since they were focusing on external characteristics. Sometimes they would make a claim about parallelograms that were not necessarily true all the time. In these cases I asked if that was always true or if they could think of a situation when that might not happen.

At the end of the 10 minutes, I had students share out what they found:

1. The opposite sides of a parallelogram are parallel
2. The opposite interior angles are congruent
3. The opposite sides are congruent

After the last two statements I wrote, "why?"

I then referenced the quadrilateral properties grid that I passed out last class. We worked through the parallelogram column. From the three statements posed above, we were able to work through all but the last three characteristics, which all dealt with diagonal properties. These properties deal with diagonals being perpendicular, bisecting each other, or being congruent.

I asked the class how they could tell if two line segments or lines were perpendicular. No one remembered. I told them that we would need to use slope. Both of these topics are from Algebra I, so I know they have been taught these ideas. I briefly went through a few ideas, such as slopes of perpendicular lines are negative reciprocals, slopes of parallel lines are equal, and how slope measures relative changes in height (y) versus length (x). I used mountain slopes and skateboard ramp steepness to help connect these ideas.

I then asked students to calculate the slope for one pair of parallel lines that they drew in order to determine if, in fact, the drawing they made was a parallelogram. With a bit of assistance here and there, most students seemed to recall how to do this.

For those students that didn't remember or struggled, I encouraged them to go to Khan Academy to review and practice, as I know there are lessons on calculating slopes, as well as working with slopes of parallel and perpendicular lines.

Next class we'll focus on the diagonals of parallelograms and their properties. We'll be calculating slopes, midpoints, and lengths for these, so we'll have some good algebra review, which may prove to be a struggle for some.