When do parallelograms have congruent diagonals or perpendicular diagonals?

Today we continued our investigation of parallelogram diagonals. Specifically, we were trying to answer two questions:

1. Do parallelograms with congruent diagonals share any common characteristics?
2. Do parallelograms with perpendicular diagonals share any common characteristics?

These questions were driven by the fact that sometimes parallelograms have perpendicular diagonals and sometimes they have congruent diagonals.

Students used whiteboard grids or graph paper to help them investigate these situations. I instructed them to work visually and use rulers and protractors in their search. Once they thought they had a parallelogram that met one or both of the conditions, they were to verify through mathematics that their figure worked. To do this they would need to use the distance formula to calculate diagonal lengths and calculate slopes of diagonals to determine if they were negative reciprocals. In addition, I told the class they would need to calculate the diagonal midpoints to check that the diagonals bisected each other, which would confirm that their figure was actually a parallelogram.

I walked around to check on progress. A few students needed some additional direction but most jumped in to tackle the job. A couple of times I had to ask students what they were seeing: Did the diagonals look congruent? Did the diagonals look perpendicular? I then asked these students how their parallelogram could be altered to get closer to either of these goals.

Toward the end of class, we were able to share that rectangles and squares had congruent diagonals and that squares and rhombuses had perpendicular diagonals. Since not every student had reached the same point, I assigned the task of verifying these results through example as homework.

I wanted to check on understanding and asked students what the official definition of a rectangle was and what the official definition of a rhombus was. I was pleased when the definitions offered up began with "A rectangle is a parallelogram that has..." or "A rhombus is a parallelogram that has..."

We'll finish up the property grids for rhombuses and rectangles next class. My intent is to work through some practice problems and then work through an investigation that will allow me to assess how well the class understands their work with parallelograms.