Rectangles, rhombuses, and squares

This class started with confirming properties for rectangles, rhombuses, and squares. Students were able to confirm at home that rectangles had congruent diagonals and that rhombuses had perpendicular diagonals. We updated the quadrilateral property grid with information for rectangles, rhombuses, and squares. (See the Jan. 6th post for information on the quadrilateral property grid.)

At this point, a student asked what the official definition of a rhombus was. I asked the class to look at their grid and to identify how rhombuses differ from parallelograms. There are two characteristics that parallelograms sometimes have and rhombuses always have: 1) all sides are congruent and 2) diagonals are perpendicular. I told the class that you could define a rhombus based on either of these two characteristics and then demonstrate (prove) that the other property had to be true. It's easier to think of rhombuses in terms of side lengths than in terms of perpendicular diagonals, so that is how a rhombus is defined: A rhombus is a parallelogram that has all sides congruent.

Of course, this definition requires knowing the definition of a parallelogram, etc. There is a large amount of layering that goes into mathematical definitions as we progress through the curriculum and I'm not sure that students always have that foundation to really understand the depth and intricacies for these definitions. That is where building conceptual knowledge helps to bridge the gap and make the definitions more understandable.

With the grid complete, I wanted students to work through a series of investigative problems and activities. As mentioned previously, I am drawing resources from the Simi Valley Unified School District Common Core Mathematics geometry site.

I had students work through Parallelogram Conjectures and Proofs. This work allows students to express reasoning, make use of properties that have just learned, revisit triangle congruence theorems as part of their justifications, work with angle relationships formed by intersecting lines, and explore connections to congruence under rigid motions.

Most students were able to get through the first ten tasks. I left the remainder as homework. Students still struggled with expressing their reasoning but are getting a little bit better. Many needed to be reminded about looking for congruent triangles to then prove that corresponding parts were congruent. Once they got through the first task using this idea they were able to proceed through the others without much assistance.

We'll continue working through these tasks. The plan is to use Guess My Parallelogram next class.