Darphin - Exploring Mathematical Reasoning - Area & Perimeter

The relationship between area and perimeter could be explored with scratch if students imagined themselves as competing real estate developers.  I imagine students creating a game where you build houses.  Students could choose to make this game multiple players or not.  Students could propose various goals, for example build houses with the most area (house would sell for more) or built the most amount of houses (more sales). Posing limitations on players resources could cause exploration of proportional and nonproportional relationships.  Below are just a few that came to mind.

• Say players must purchase building blocks for perimeter - students become conscious of how perimeter affects area.  
• Requiring each building built to have the same perimeter could lead students to explore different shapes and which shapes use perimeter more efficiently in terms of area.
• I would assume students would also find shapes like circle pose other limitations on the game like loss of available space for other buildings (if players were not allowed to create irregularly shaped buildings).
• If students could only create squares and earned money (points) for area but had to spend money (lost points) for perimeter they could find that area and side length is not a proportional relationship for squares.
• Limitation on time to build a building could help students develop concrete mental “rules” about these relationships. The players would need to have a grasp the relationship implications between perimeter and area quickly when it was their turn to build.

Assessment

When thinking of a math concept to be explored, I tried to consider, hopefully somewhat successfully, principles Papert discussed in An Exploration in the Space of Mathematics Education.  Grover suggested “systems of assessments” to tackle the assessment concern with this type of learning.  I did not find these “systems of assessments” as drastically different from how teachers already assess student learning in elementary settings.

Considering my proposal with area and perimeter, a pre- and post-test would be useful for gauging concepts developed.  Observing students play the game and narrating their choices could demonstrate relationship reasoning developed.