We continued looking at diagonals in parallelograms. Students didn't do well over the weekend in finding midpoints and segment lengths, so I got the coordinates for the endpoints of a diagonal for one students and quickly ran through how to find a midpoint and the length. After a few clarifying questions, I released the class to find the lengths and midpoints for the diagonals of their parallelograms. Many students drew new figures and most were hesitant or struggling with the calculations, so this took a lot longer than I had hoped. By the end of class, students were successful in computing their midpoints and their diagonal lengths. We reviewed the results and concluded that the diagonals of parallelograms always bisect each other. We also concluded that the diagonals are sometimes congruent.
At this point we have found for parallelograms that:
- Opposite sides are always parallel
- Opposite sides are always congruent
- Opposite angles are always congruent
- Diagonals always bisect each other
- Diagonals sometimes are congruent
- Diagonals sometimes are perpendicular
- All sides are sometimes congruent
- All angles are sometimes congruent
For the next class I intend to have students explore what types of parallelograms have perpendicular diagonals and what types of parallelograms have congruent diagonals. I'll use this to connect to when all sides are sometimes congruent and when all angles are sometimes congruent to flesh out the properties of rectangles and rhombuses.
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