Polygon congruence criteria
Remember triangle congruence criteria? SSS, SAS, ASA, AAS? Those are foundations of congruence in Geometry...so what do we need to know in order to determine if two quadrilaterals are congruent? Or two pentagons? Hexagons? Or ANY n-sided shape? Triangle congruence criteria is well-documented - it's been discussed in math classes for a very long time. Quadrilateral congruence is established as well - the Geometry students have used some of the concepts from this article to help build on some of these ideas. Beyond quadrilateral congruence criteria though, there isn't very much that is known about what criteria (and how many pieces of information you need to know!) it takes to prove that two n-gons are congruent...so that's exactly what the Geometry students are trying to do! For the past several weeks, students have trying to find out how many pieces of information you need to know in order to prove that two n-gons are congruent and what general patterns of congruence there are.
These students are exploring areas of geometry that either have not been explored before or at least are not well-known enough to be easily accessible. Students are eventually hoping to write up their findings and submit them to a peer-reviewed journal for publishing.
Students started with investigating quadrilateral congruence criteria and trying to discover which sets of criteria worked by making lists, experimenting with manipulatives, and testing for counter-examples.
These students are exploring areas of geometry that either have not been explored before or at least are not well-known enough to be easily accessible. Students are eventually hoping to write up their findings and submit them to a peer-reviewed journal for publishing.
Students started with investigating quadrilateral congruence criteria and trying to discover which sets of criteria worked by making lists, experimenting with manipulatives, and testing for counter-examples.
Once they had the criteria for the quadrilaterals, they moved on to pentagons. They started by organizing the different combinations:
and then attempting to come up with counter examples
The classes kept track of their ideas by making class lists of congruence conditions that worked and ones that didn't.
We're currently working on finding and extending patterns into polygons with a much larger number of sides, attempting to PROVE these congruence conditions for polygons of any size, and then writing up a formal mathematical paper. Stay tuned!
Students are currently deciding on how to discuss and showcase their findings. Check back when they've decided on a final product!